Abstract

A nanophotonic trapping platform based on on-chip tunable optical interference allows parallel processing of biomolecules and holds promise to make single molecule manipulation and precision measurements more easily and broadly available. The nanophotonic standing wave array trap (nSWAT) device [Nat. Nanotechnol. 9, 448 (2014); Nano Lett. 16, 6661 (2016)] represents such a platform and can trap a large array of beads by the evanescent field of the standing wave of a nanophotonic waveguide and reposition them using an integrated microheater. In this paper, by taking a systematic design approach, we present a new generation of nSWAT devices with significant enhancement of the optical trapping force, stiffness, and stability, while the quality of the standing wave trap is resistant to fabrication imperfections. The device is implemented on a silicon nitride photonic platform and operates at 1064 nm wavelength which permits low optical absorption by the aqueous solution. Such performance improvements open a broader range of applications based on these on-chip optical traps.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref]
  2. K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
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  3. S. Forth, M. Y. Sheinin, J. Inman, and M. D. Wang, “Torque Measurement at the Single-Molecule Level,” Annu. Rev. Biophys. 42(1), 583–604 (2013).
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  5. J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
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  6. D. S. Johnson, L. Bai, B. Y. Smith, S. S. Patel, and M. D. Wang, “Single-molecule studies reveal dynamics of DNA unwinding by the ring-shaped T7 helicase,” Cell 129(7), 1299–1309 (2007).
    [Crossref] [PubMed]
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  11. J. Ma, L. Bai, and M. D. Wang, “Transcription under torsion,” Science 340(6140), 1580–1583 (2013).
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    [Crossref] [PubMed]
  13. J. E. Molloy, J. E. Burns, J. Kendrick-Jones, R. T. Tregear, and D. C. S. White, “Movement and Force Produced by a Single Myosin Head,” Nature 378(6553), 209–212 (1995).
    [Crossref] [PubMed]
  14. M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nat. Phys. 3(7), 477–480 (2007).
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  22. F. Ye, R. P. Badman, J. T. Inman, M. Soltani, J. L. Killian, and M. D. Wang, “Biocompatible and high stiffness nanophotonic trap array for precise and versatile manipulation,” Nano Lett. 16(10), 6661–6667 (2016).
    [Crossref] [PubMed]
  23. S. Gaugiran, S. Gétin, J. Fedeli, G. Colas, A. Fuchs, F. Chatelain, and J. Dérouard, “Optical manipulation of microparticles and cells on silicon nitride waveguides,” Opt. Express 13(18), 6956–6963 (2005).
    [Crossref] [PubMed]
  24. M. Soltani, J. T. Inman, M. Lipson, and M. D. Wang, “Electro-optofluidics: achieving dynamic control on-chip,” Opt. Express 20(20), 22314–22326 (2012).
    [Crossref] [PubMed]
  25. F. P. Payne and J. P. R. Lacey, “A theoretical-analysis of scattering loss from planar optical wave-guides,” Opt. Quantum Electron. 26(10), 977–986 (1994).
    [Crossref]
  26. M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72(3), 1335–1346 (1997).
    [Crossref] [PubMed]
  27. A. La Porta and M. D. Wang, “Optical torque wrench: angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett. 92(19), 190801 (2004).
    [Crossref] [PubMed]
  28. C. Deufel and M. D. Wang, “Detection of forces and displacements along the axial direction in an optical trap,” Biophys. J. 90(2), 657–667 (2006).
    [Crossref] [PubMed]

2016 (1)

F. Ye, R. P. Badman, J. T. Inman, M. Soltani, J. L. Killian, and M. D. Wang, “Biocompatible and high stiffness nanophotonic trap array for precise and versatile manipulation,” Nano Lett. 16(10), 6661–6667 (2016).
[Crossref] [PubMed]

2014 (1)

M. Soltani, J. Lin, R. A. Forties, J. T. Inman, S. N. Saraf, R. M. Fulbright, M. Lipson, and M. D. Wang, “Nanophotonic trapping for precise manipulation of biomolecular arrays,” Nat. Nanotechnol. 9(6), 448–452 (2014).
[Crossref] [PubMed]

2013 (2)

S. Forth, M. Y. Sheinin, J. Inman, and M. D. Wang, “Torque Measurement at the Single-Molecule Level,” Annu. Rev. Biophys. 42(1), 583–604 (2013).
[Crossref] [PubMed]

J. Ma, L. Bai, and M. D. Wang, “Transcription under torsion,” Science 340(6140), 1580–1583 (2013).
[Crossref] [PubMed]

2012 (1)

2011 (5)

M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photonics 5(6), 349–356 (2011).
[Crossref]

K. Wang, E. Schonbrun, P. Steinvurzel, and K. B. Crozier, “Trapping and rotating nanoparticles using a plasmonic nano-tweezer with an integrated heat sink,” Nat. Commun. 2, 469 (2011).
[Crossref] [PubMed]

D. Erickson, X. Serey, Y. F. Chen, and S. Mandal, “Nanomanipulation using near field photonics,” Lab Chip 11(6), 995–1009 (2011).
[Crossref] [PubMed]

B. Sun, D. S. Johnson, G. Patel, B. Y. Smith, M. Pandey, S. S. Patel, and M. D. Wang, “ATP-induced helicase slippage reveals highly coordinated subunits,” Nature 478(7367), 132–135 (2011).
[Crossref] [PubMed]

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[Crossref]

2010 (4)

J. Hilario and S. C. Kowalczykowski, “Visualizing protein-DNA interactions at the single-molecule level,” Curr. Opin. Chem. Biol. 14(1), 15–22 (2010).
[Crossref] [PubMed]

A. Farré, A. van der Horst, G. A. Blab, B. P. B. Downing, and N. R. Forde, “Stretching single DNA molecules to demonstrate high-force capabilities of holographic optical tweezers,” J. Biophotonics 3(4), 224–233 (2010).
[Crossref] [PubMed]

S. Mandal, X. Serey, and D. Erickson, “Nanomanipulation Using Silicon Photonic Crystal Resonators,” Nano Lett. 10(1), 99–104 (2010).
[Crossref] [PubMed]

S. Lin, E. Schonbrun, and K. Crozier, “Optical manipulation with planar silicon microring resonators,” Nano Lett. 10(7), 2408–2411 (2010).
[Crossref] [PubMed]

2009 (2)

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]

M. A. Hall, A. Shundrovsky, L. Bai, R. M. Fulbright, J. T. Lis, and M. D. Wang, “High-resolution dynamic mapping of histone-DNA interactions in a nucleosome,” Nat. Struct. Mol. Biol. 16(2), 124–129 (2009).
[Crossref] [PubMed]

2008 (2)

K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods 5(6), 491–505 (2008).
[Crossref] [PubMed]

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[Crossref] [PubMed]

2007 (2)

D. S. Johnson, L. Bai, B. Y. Smith, S. S. Patel, and M. D. Wang, “Single-molecule studies reveal dynamics of DNA unwinding by the ring-shaped T7 helicase,” Cell 129(7), 1299–1309 (2007).
[Crossref] [PubMed]

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nat. Phys. 3(7), 477–480 (2007).
[Crossref]

2006 (1)

C. Deufel and M. D. Wang, “Detection of forces and displacements along the axial direction in an optical trap,” Biophys. J. 90(2), 657–667 (2006).
[Crossref] [PubMed]

2005 (1)

2004 (1)

A. La Porta and M. D. Wang, “Optical torque wrench: angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett. 92(19), 190801 (2004).
[Crossref] [PubMed]

2003 (1)

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

1998 (1)

S. Chu, “The manipulation of neutral particles,” Rev. Mod. Phys. 70(3), 685–706 (1998).
[Crossref]

1997 (1)

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72(3), 1335–1346 (1997).
[Crossref] [PubMed]

1995 (1)

J. E. Molloy, J. E. Burns, J. Kendrick-Jones, R. T. Tregear, and D. C. S. White, “Movement and Force Produced by a Single Myosin Head,” Nature 378(6553), 209–212 (1995).
[Crossref] [PubMed]

1994 (1)

F. P. Payne and J. P. R. Lacey, “A theoretical-analysis of scattering loss from planar optical wave-guides,” Opt. Quantum Electron. 26(10), 977–986 (1994).
[Crossref]

Badman, R. P.

F. Ye, R. P. Badman, J. T. Inman, M. Soltani, J. L. Killian, and M. D. Wang, “Biocompatible and high stiffness nanophotonic trap array for precise and versatile manipulation,” Nano Lett. 16(10), 6661–6667 (2016).
[Crossref] [PubMed]

Bai, L.

J. Ma, L. Bai, and M. D. Wang, “Transcription under torsion,” Science 340(6140), 1580–1583 (2013).
[Crossref] [PubMed]

M. A. Hall, A. Shundrovsky, L. Bai, R. M. Fulbright, J. T. Lis, and M. D. Wang, “High-resolution dynamic mapping of histone-DNA interactions in a nucleosome,” Nat. Struct. Mol. Biol. 16(2), 124–129 (2009).
[Crossref] [PubMed]

D. S. Johnson, L. Bai, B. Y. Smith, S. S. Patel, and M. D. Wang, “Single-molecule studies reveal dynamics of DNA unwinding by the ring-shaped T7 helicase,” Cell 129(7), 1299–1309 (2007).
[Crossref] [PubMed]

Blab, G. A.

A. Farré, A. van der Horst, G. A. Blab, B. P. B. Downing, and N. R. Forde, “Stretching single DNA molecules to demonstrate high-force capabilities of holographic optical tweezers,” J. Biophotonics 3(4), 224–233 (2010).
[Crossref] [PubMed]

Block, S. M.

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72(3), 1335–1346 (1997).
[Crossref] [PubMed]

Burns, J. E.

J. E. Molloy, J. E. Burns, J. Kendrick-Jones, R. T. Tregear, and D. C. S. White, “Movement and Force Produced by a Single Myosin Head,” Nature 378(6553), 209–212 (1995).
[Crossref] [PubMed]

Bustamante, C.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[Crossref] [PubMed]

Chatelain, F.

Chemla, Y. R.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[Crossref] [PubMed]

Chen, Y. F.

D. Erickson, X. Serey, Y. F. Chen, and S. Mandal, “Nanomanipulation using near field photonics,” Lab Chip 11(6), 995–1009 (2011).
[Crossref] [PubMed]

Chu, S.

S. Chu, “The manipulation of neutral particles,” Rev. Mod. Phys. 70(3), 685–706 (1998).
[Crossref]

Cizmar, T.

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[Crossref]

Colas, G.

Crozier, K.

S. Lin, E. Schonbrun, and K. Crozier, “Optical manipulation with planar silicon microring resonators,” Nano Lett. 10(7), 2408–2411 (2010).
[Crossref] [PubMed]

Crozier, K. B.

K. Wang, E. Schonbrun, P. Steinvurzel, and K. B. Crozier, “Trapping and rotating nanoparticles using a plasmonic nano-tweezer with an integrated heat sink,” Nat. Commun. 2, 469 (2011).
[Crossref] [PubMed]

Dérouard, J.

Deufel, C.

C. Deufel and M. D. Wang, “Detection of forces and displacements along the axial direction in an optical trap,” Biophys. J. 90(2), 657–667 (2006).
[Crossref] [PubMed]

Dholakia, K.

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[Crossref]

Downing, B. P. B.

A. Farré, A. van der Horst, G. A. Blab, B. P. B. Downing, and N. R. Forde, “Stretching single DNA molecules to demonstrate high-force capabilities of holographic optical tweezers,” J. Biophotonics 3(4), 224–233 (2010).
[Crossref] [PubMed]

Erickson, D.

D. Erickson, X. Serey, Y. F. Chen, and S. Mandal, “Nanomanipulation using near field photonics,” Lab Chip 11(6), 995–1009 (2011).
[Crossref] [PubMed]

S. Mandal, X. Serey, and D. Erickson, “Nanomanipulation Using Silicon Photonic Crystal Resonators,” Nano Lett. 10(1), 99–104 (2010).
[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]

Farré, A.

A. Farré, A. van der Horst, G. A. Blab, B. P. B. Downing, and N. R. Forde, “Stretching single DNA molecules to demonstrate high-force capabilities of holographic optical tweezers,” J. Biophotonics 3(4), 224–233 (2010).
[Crossref] [PubMed]

Fedeli, J.

Forde, N. R.

A. Farré, A. van der Horst, G. A. Blab, B. P. B. Downing, and N. R. Forde, “Stretching single DNA molecules to demonstrate high-force capabilities of holographic optical tweezers,” J. Biophotonics 3(4), 224–233 (2010).
[Crossref] [PubMed]

Forth, S.

S. Forth, M. Y. Sheinin, J. Inman, and M. D. Wang, “Torque Measurement at the Single-Molecule Level,” Annu. Rev. Biophys. 42(1), 583–604 (2013).
[Crossref] [PubMed]

Forties, R. A.

M. Soltani, J. Lin, R. A. Forties, J. T. Inman, S. N. Saraf, R. M. Fulbright, M. Lipson, and M. D. Wang, “Nanophotonic trapping for precise manipulation of biomolecular arrays,” Nat. Nanotechnol. 9(6), 448–452 (2014).
[Crossref] [PubMed]

Fuchs, A.

Fulbright, R. M.

M. Soltani, J. Lin, R. A. Forties, J. T. Inman, S. N. Saraf, R. M. Fulbright, M. Lipson, and M. D. Wang, “Nanophotonic trapping for precise manipulation of biomolecular arrays,” Nat. Nanotechnol. 9(6), 448–452 (2014).
[Crossref] [PubMed]

M. A. Hall, A. Shundrovsky, L. Bai, R. M. Fulbright, J. T. Lis, and M. D. Wang, “High-resolution dynamic mapping of histone-DNA interactions in a nucleosome,” Nat. Struct. Mol. Biol. 16(2), 124–129 (2009).
[Crossref] [PubMed]

Gaugiran, S.

Gelles, J.

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72(3), 1335–1346 (1997).
[Crossref] [PubMed]

Gétin, S.

Girard, C.

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nat. Phys. 3(7), 477–480 (2007).
[Crossref]

Grier, D. G.

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

Hall, M. A.

M. A. Hall, A. Shundrovsky, L. Bai, R. M. Fulbright, J. T. Lis, and M. D. Wang, “High-resolution dynamic mapping of histone-DNA interactions in a nucleosome,” Nat. Struct. Mol. Biol. 16(2), 124–129 (2009).
[Crossref] [PubMed]

Hilario, J.

J. Hilario and S. C. Kowalczykowski, “Visualizing protein-DNA interactions at the single-molecule level,” Curr. Opin. Chem. Biol. 14(1), 15–22 (2010).
[Crossref] [PubMed]

Inman, J.

S. Forth, M. Y. Sheinin, J. Inman, and M. D. Wang, “Torque Measurement at the Single-Molecule Level,” Annu. Rev. Biophys. 42(1), 583–604 (2013).
[Crossref] [PubMed]

Inman, J. T.

F. Ye, R. P. Badman, J. T. Inman, M. Soltani, J. L. Killian, and M. D. Wang, “Biocompatible and high stiffness nanophotonic trap array for precise and versatile manipulation,” Nano Lett. 16(10), 6661–6667 (2016).
[Crossref] [PubMed]

M. Soltani, J. Lin, R. A. Forties, J. T. Inman, S. N. Saraf, R. M. Fulbright, M. Lipson, and M. D. Wang, “Nanophotonic trapping for precise manipulation of biomolecular arrays,” Nat. Nanotechnol. 9(6), 448–452 (2014).
[Crossref] [PubMed]

M. Soltani, J. T. Inman, M. Lipson, and M. D. Wang, “Electro-optofluidics: achieving dynamic control on-chip,” Opt. Express 20(20), 22314–22326 (2012).
[Crossref] [PubMed]

Johnson, D. S.

B. Sun, D. S. Johnson, G. Patel, B. Y. Smith, M. Pandey, S. S. Patel, and M. D. Wang, “ATP-induced helicase slippage reveals highly coordinated subunits,” Nature 478(7367), 132–135 (2011).
[Crossref] [PubMed]

D. S. Johnson, L. Bai, B. Y. Smith, S. S. Patel, and M. D. Wang, “Single-molecule studies reveal dynamics of DNA unwinding by the ring-shaped T7 helicase,” Cell 129(7), 1299–1309 (2007).
[Crossref] [PubMed]

Juan, M. L.

M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photonics 5(6), 349–356 (2011).
[Crossref]

Kendrick-Jones, J.

J. E. Molloy, J. E. Burns, J. Kendrick-Jones, R. T. Tregear, and D. C. S. White, “Movement and Force Produced by a Single Myosin Head,” Nature 378(6553), 209–212 (1995).
[Crossref] [PubMed]

Killian, J. L.

F. Ye, R. P. Badman, J. T. Inman, M. Soltani, J. L. Killian, and M. D. Wang, “Biocompatible and high stiffness nanophotonic trap array for precise and versatile manipulation,” Nano Lett. 16(10), 6661–6667 (2016).
[Crossref] [PubMed]

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]

Kowalczykowski, S. C.

J. Hilario and S. C. Kowalczykowski, “Visualizing protein-DNA interactions at the single-molecule level,” Curr. Opin. Chem. Biol. 14(1), 15–22 (2010).
[Crossref] [PubMed]

La Porta, A.

A. La Porta and M. D. Wang, “Optical torque wrench: angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett. 92(19), 190801 (2004).
[Crossref] [PubMed]

Lacey, J. P. R.

F. P. Payne and J. P. R. Lacey, “A theoretical-analysis of scattering loss from planar optical wave-guides,” Opt. Quantum Electron. 26(10), 977–986 (1994).
[Crossref]

Landick, R.

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72(3), 1335–1346 (1997).
[Crossref] [PubMed]

Lin, J.

M. Soltani, J. Lin, R. A. Forties, J. T. Inman, S. N. Saraf, R. M. Fulbright, M. Lipson, and M. D. Wang, “Nanophotonic trapping for precise manipulation of biomolecular arrays,” Nat. Nanotechnol. 9(6), 448–452 (2014).
[Crossref] [PubMed]

Lin, S.

S. Lin, E. Schonbrun, and K. Crozier, “Optical manipulation with planar silicon microring resonators,” Nano Lett. 10(7), 2408–2411 (2010).
[Crossref] [PubMed]

Lipson, M.

M. Soltani, J. Lin, R. A. Forties, J. T. Inman, S. N. Saraf, R. M. Fulbright, M. Lipson, and M. D. Wang, “Nanophotonic trapping for precise manipulation of biomolecular arrays,” Nat. Nanotechnol. 9(6), 448–452 (2014).
[Crossref] [PubMed]

M. Soltani, J. T. Inman, M. Lipson, and M. D. Wang, “Electro-optofluidics: achieving dynamic control on-chip,” Opt. Express 20(20), 22314–22326 (2012).
[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]

Lis, J. T.

M. A. Hall, A. Shundrovsky, L. Bai, R. M. Fulbright, J. T. Lis, and M. D. Wang, “High-resolution dynamic mapping of histone-DNA interactions in a nucleosome,” Nat. Struct. Mol. Biol. 16(2), 124–129 (2009).
[Crossref] [PubMed]

Ma, J.

J. Ma, L. Bai, and M. D. Wang, “Transcription under torsion,” Science 340(6140), 1580–1583 (2013).
[Crossref] [PubMed]

Mandal, S.

D. Erickson, X. Serey, Y. F. Chen, and S. Mandal, “Nanomanipulation using near field photonics,” Lab Chip 11(6), 995–1009 (2011).
[Crossref] [PubMed]

S. Mandal, X. Serey, and D. Erickson, “Nanomanipulation Using Silicon Photonic Crystal Resonators,” Nano Lett. 10(1), 99–104 (2010).
[Crossref] [PubMed]

Moffitt, J. R.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[Crossref] [PubMed]

Molloy, J. E.

J. E. Molloy, J. E. Burns, J. Kendrick-Jones, R. T. Tregear, and D. C. S. White, “Movement and Force Produced by a Single Myosin Head,” Nature 378(6553), 209–212 (1995).
[Crossref] [PubMed]

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]

Nagy, A.

K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods 5(6), 491–505 (2008).
[Crossref] [PubMed]

Neuman, K. C.

K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods 5(6), 491–505 (2008).
[Crossref] [PubMed]

Pandey, M.

B. Sun, D. S. Johnson, G. Patel, B. Y. Smith, M. Pandey, S. S. Patel, and M. D. Wang, “ATP-induced helicase slippage reveals highly coordinated subunits,” Nature 478(7367), 132–135 (2011).
[Crossref] [PubMed]

Patel, G.

B. Sun, D. S. Johnson, G. Patel, B. Y. Smith, M. Pandey, S. S. Patel, and M. D. Wang, “ATP-induced helicase slippage reveals highly coordinated subunits,” Nature 478(7367), 132–135 (2011).
[Crossref] [PubMed]

Patel, S. S.

B. Sun, D. S. Johnson, G. Patel, B. Y. Smith, M. Pandey, S. S. Patel, and M. D. Wang, “ATP-induced helicase slippage reveals highly coordinated subunits,” Nature 478(7367), 132–135 (2011).
[Crossref] [PubMed]

D. S. Johnson, L. Bai, B. Y. Smith, S. S. Patel, and M. D. Wang, “Single-molecule studies reveal dynamics of DNA unwinding by the ring-shaped T7 helicase,” Cell 129(7), 1299–1309 (2007).
[Crossref] [PubMed]

Payne, F. P.

F. P. Payne and J. P. R. Lacey, “A theoretical-analysis of scattering loss from planar optical wave-guides,” Opt. Quantum Electron. 26(10), 977–986 (1994).
[Crossref]

Quidant, R.

M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photonics 5(6), 349–356 (2011).
[Crossref]

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nat. Phys. 3(7), 477–480 (2007).
[Crossref]

Righini, M.

M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photonics 5(6), 349–356 (2011).
[Crossref]

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nat. Phys. 3(7), 477–480 (2007).
[Crossref]

Saraf, S. N.

M. Soltani, J. Lin, R. A. Forties, J. T. Inman, S. N. Saraf, R. M. Fulbright, M. Lipson, and M. D. Wang, “Nanophotonic trapping for precise manipulation of biomolecular arrays,” Nat. Nanotechnol. 9(6), 448–452 (2014).
[Crossref] [PubMed]

Schmidt, B. S.

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]

Schonbrun, E.

K. Wang, E. Schonbrun, P. Steinvurzel, and K. B. Crozier, “Trapping and rotating nanoparticles using a plasmonic nano-tweezer with an integrated heat sink,” Nat. Commun. 2, 469 (2011).
[Crossref] [PubMed]

S. Lin, E. Schonbrun, and K. Crozier, “Optical manipulation with planar silicon microring resonators,” Nano Lett. 10(7), 2408–2411 (2010).
[Crossref] [PubMed]

Serey, X.

D. Erickson, X. Serey, Y. F. Chen, and S. Mandal, “Nanomanipulation using near field photonics,” Lab Chip 11(6), 995–1009 (2011).
[Crossref] [PubMed]

S. Mandal, X. Serey, and D. Erickson, “Nanomanipulation Using Silicon Photonic Crystal Resonators,” Nano Lett. 10(1), 99–104 (2010).
[Crossref] [PubMed]

Sheinin, M. Y.

S. Forth, M. Y. Sheinin, J. Inman, and M. D. Wang, “Torque Measurement at the Single-Molecule Level,” Annu. Rev. Biophys. 42(1), 583–604 (2013).
[Crossref] [PubMed]

Shundrovsky, A.

M. A. Hall, A. Shundrovsky, L. Bai, R. M. Fulbright, J. T. Lis, and M. D. Wang, “High-resolution dynamic mapping of histone-DNA interactions in a nucleosome,” Nat. Struct. Mol. Biol. 16(2), 124–129 (2009).
[Crossref] [PubMed]

Smith, B. Y.

B. Sun, D. S. Johnson, G. Patel, B. Y. Smith, M. Pandey, S. S. Patel, and M. D. Wang, “ATP-induced helicase slippage reveals highly coordinated subunits,” Nature 478(7367), 132–135 (2011).
[Crossref] [PubMed]

D. S. Johnson, L. Bai, B. Y. Smith, S. S. Patel, and M. D. Wang, “Single-molecule studies reveal dynamics of DNA unwinding by the ring-shaped T7 helicase,” Cell 129(7), 1299–1309 (2007).
[Crossref] [PubMed]

Smith, S. B.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[Crossref] [PubMed]

Soltani, M.

F. Ye, R. P. Badman, J. T. Inman, M. Soltani, J. L. Killian, and M. D. Wang, “Biocompatible and high stiffness nanophotonic trap array for precise and versatile manipulation,” Nano Lett. 16(10), 6661–6667 (2016).
[Crossref] [PubMed]

M. Soltani, J. Lin, R. A. Forties, J. T. Inman, S. N. Saraf, R. M. Fulbright, M. Lipson, and M. D. Wang, “Nanophotonic trapping for precise manipulation of biomolecular arrays,” Nat. Nanotechnol. 9(6), 448–452 (2014).
[Crossref] [PubMed]

M. Soltani, J. T. Inman, M. Lipson, and M. D. Wang, “Electro-optofluidics: achieving dynamic control on-chip,” Opt. Express 20(20), 22314–22326 (2012).
[Crossref] [PubMed]

Steinvurzel, P.

K. Wang, E. Schonbrun, P. Steinvurzel, and K. B. Crozier, “Trapping and rotating nanoparticles using a plasmonic nano-tweezer with an integrated heat sink,” Nat. Commun. 2, 469 (2011).
[Crossref] [PubMed]

Sun, B.

B. Sun, D. S. Johnson, G. Patel, B. Y. Smith, M. Pandey, S. S. Patel, and M. D. Wang, “ATP-induced helicase slippage reveals highly coordinated subunits,” Nature 478(7367), 132–135 (2011).
[Crossref] [PubMed]

Tregear, R. T.

J. E. Molloy, J. E. Burns, J. Kendrick-Jones, R. T. Tregear, and D. C. S. White, “Movement and Force Produced by a Single Myosin Head,” Nature 378(6553), 209–212 (1995).
[Crossref] [PubMed]

van der Horst, A.

A. Farré, A. van der Horst, G. A. Blab, B. P. B. Downing, and N. R. Forde, “Stretching single DNA molecules to demonstrate high-force capabilities of holographic optical tweezers,” J. Biophotonics 3(4), 224–233 (2010).
[Crossref] [PubMed]

Wang, K.

K. Wang, E. Schonbrun, P. Steinvurzel, and K. B. Crozier, “Trapping and rotating nanoparticles using a plasmonic nano-tweezer with an integrated heat sink,” Nat. Commun. 2, 469 (2011).
[Crossref] [PubMed]

Wang, M. D.

F. Ye, R. P. Badman, J. T. Inman, M. Soltani, J. L. Killian, and M. D. Wang, “Biocompatible and high stiffness nanophotonic trap array for precise and versatile manipulation,” Nano Lett. 16(10), 6661–6667 (2016).
[Crossref] [PubMed]

M. Soltani, J. Lin, R. A. Forties, J. T. Inman, S. N. Saraf, R. M. Fulbright, M. Lipson, and M. D. Wang, “Nanophotonic trapping for precise manipulation of biomolecular arrays,” Nat. Nanotechnol. 9(6), 448–452 (2014).
[Crossref] [PubMed]

J. Ma, L. Bai, and M. D. Wang, “Transcription under torsion,” Science 340(6140), 1580–1583 (2013).
[Crossref] [PubMed]

S. Forth, M. Y. Sheinin, J. Inman, and M. D. Wang, “Torque Measurement at the Single-Molecule Level,” Annu. Rev. Biophys. 42(1), 583–604 (2013).
[Crossref] [PubMed]

M. Soltani, J. T. Inman, M. Lipson, and M. D. Wang, “Electro-optofluidics: achieving dynamic control on-chip,” Opt. Express 20(20), 22314–22326 (2012).
[Crossref] [PubMed]

B. Sun, D. S. Johnson, G. Patel, B. Y. Smith, M. Pandey, S. S. Patel, and M. D. Wang, “ATP-induced helicase slippage reveals highly coordinated subunits,” Nature 478(7367), 132–135 (2011).
[Crossref] [PubMed]

M. A. Hall, A. Shundrovsky, L. Bai, R. M. Fulbright, J. T. Lis, and M. D. Wang, “High-resolution dynamic mapping of histone-DNA interactions in a nucleosome,” Nat. Struct. Mol. Biol. 16(2), 124–129 (2009).
[Crossref] [PubMed]

D. S. Johnson, L. Bai, B. Y. Smith, S. S. Patel, and M. D. Wang, “Single-molecule studies reveal dynamics of DNA unwinding by the ring-shaped T7 helicase,” Cell 129(7), 1299–1309 (2007).
[Crossref] [PubMed]

C. Deufel and M. D. Wang, “Detection of forces and displacements along the axial direction in an optical trap,” Biophys. J. 90(2), 657–667 (2006).
[Crossref] [PubMed]

A. La Porta and M. D. Wang, “Optical torque wrench: angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett. 92(19), 190801 (2004).
[Crossref] [PubMed]

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72(3), 1335–1346 (1997).
[Crossref] [PubMed]

White, D. C. S.

J. E. Molloy, J. E. Burns, J. Kendrick-Jones, R. T. Tregear, and D. C. S. White, “Movement and Force Produced by a Single Myosin Head,” Nature 378(6553), 209–212 (1995).
[Crossref] [PubMed]

Yang, A. H. J.

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]

Ye, F.

F. Ye, R. P. Badman, J. T. Inman, M. Soltani, J. L. Killian, and M. D. Wang, “Biocompatible and high stiffness nanophotonic trap array for precise and versatile manipulation,” Nano Lett. 16(10), 6661–6667 (2016).
[Crossref] [PubMed]

Yin, H.

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72(3), 1335–1346 (1997).
[Crossref] [PubMed]

Zelenina, A. S.

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nat. Phys. 3(7), 477–480 (2007).
[Crossref]

Annu. Rev. Biochem. (1)

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[Crossref] [PubMed]

Annu. Rev. Biophys. (1)

S. Forth, M. Y. Sheinin, J. Inman, and M. D. Wang, “Torque Measurement at the Single-Molecule Level,” Annu. Rev. Biophys. 42(1), 583–604 (2013).
[Crossref] [PubMed]

Biophys. J. (2)

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72(3), 1335–1346 (1997).
[Crossref] [PubMed]

C. Deufel and M. D. Wang, “Detection of forces and displacements along the axial direction in an optical trap,” Biophys. J. 90(2), 657–667 (2006).
[Crossref] [PubMed]

Cell (1)

D. S. Johnson, L. Bai, B. Y. Smith, S. S. Patel, and M. D. Wang, “Single-molecule studies reveal dynamics of DNA unwinding by the ring-shaped T7 helicase,” Cell 129(7), 1299–1309 (2007).
[Crossref] [PubMed]

Curr. Opin. Chem. Biol. (1)

J. Hilario and S. C. Kowalczykowski, “Visualizing protein-DNA interactions at the single-molecule level,” Curr. Opin. Chem. Biol. 14(1), 15–22 (2010).
[Crossref] [PubMed]

J. Biophotonics (1)

A. Farré, A. van der Horst, G. A. Blab, B. P. B. Downing, and N. R. Forde, “Stretching single DNA molecules to demonstrate high-force capabilities of holographic optical tweezers,” J. Biophotonics 3(4), 224–233 (2010).
[Crossref] [PubMed]

Lab Chip (1)

D. Erickson, X. Serey, Y. F. Chen, and S. Mandal, “Nanomanipulation using near field photonics,” Lab Chip 11(6), 995–1009 (2011).
[Crossref] [PubMed]

Nano Lett. (3)

F. Ye, R. P. Badman, J. T. Inman, M. Soltani, J. L. Killian, and M. D. Wang, “Biocompatible and high stiffness nanophotonic trap array for precise and versatile manipulation,” Nano Lett. 16(10), 6661–6667 (2016).
[Crossref] [PubMed]

S. Mandal, X. Serey, and D. Erickson, “Nanomanipulation Using Silicon Photonic Crystal Resonators,” Nano Lett. 10(1), 99–104 (2010).
[Crossref] [PubMed]

S. Lin, E. Schonbrun, and K. Crozier, “Optical manipulation with planar silicon microring resonators,” Nano Lett. 10(7), 2408–2411 (2010).
[Crossref] [PubMed]

Nat. Commun. (1)

K. Wang, E. Schonbrun, P. Steinvurzel, and K. B. Crozier, “Trapping and rotating nanoparticles using a plasmonic nano-tweezer with an integrated heat sink,” Nat. Commun. 2, 469 (2011).
[Crossref] [PubMed]

Nat. Methods (1)

K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods 5(6), 491–505 (2008).
[Crossref] [PubMed]

Nat. Nanotechnol. (1)

M. Soltani, J. Lin, R. A. Forties, J. T. Inman, S. N. Saraf, R. M. Fulbright, M. Lipson, and M. D. Wang, “Nanophotonic trapping for precise manipulation of biomolecular arrays,” Nat. Nanotechnol. 9(6), 448–452 (2014).
[Crossref] [PubMed]

Nat. Photonics (2)

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[Crossref]

M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photonics 5(6), 349–356 (2011).
[Crossref]

Nat. Phys. (1)

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nat. Phys. 3(7), 477–480 (2007).
[Crossref]

Nat. Struct. Mol. Biol. (1)

M. A. Hall, A. Shundrovsky, L. Bai, R. M. Fulbright, J. T. Lis, and M. D. Wang, “High-resolution dynamic mapping of histone-DNA interactions in a nucleosome,” Nat. Struct. Mol. Biol. 16(2), 124–129 (2009).
[Crossref] [PubMed]

Nature (4)

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

J. E. Molloy, J. E. Burns, J. Kendrick-Jones, R. T. Tregear, and D. C. S. White, “Movement and Force Produced by a Single Myosin Head,” Nature 378(6553), 209–212 (1995).
[Crossref] [PubMed]

B. Sun, D. S. Johnson, G. Patel, B. Y. Smith, M. Pandey, S. S. Patel, and M. D. Wang, “ATP-induced helicase slippage reveals highly coordinated subunits,” Nature 478(7367), 132–135 (2011).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Quantum Electron. (1)

F. P. Payne and J. P. R. Lacey, “A theoretical-analysis of scattering loss from planar optical wave-guides,” Opt. Quantum Electron. 26(10), 977–986 (1994).
[Crossref]

Phys. Rev. Lett. (1)

A. La Porta and M. D. Wang, “Optical torque wrench: angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett. 92(19), 190801 (2004).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

S. Chu, “The manipulation of neutral particles,” Rev. Mod. Phys. 70(3), 685–706 (1998).
[Crossref]

Science (1)

J. Ma, L. Bai, and M. D. Wang, “Transcription under torsion,” Science 340(6140), 1580–1583 (2013).
[Crossref] [PubMed]

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Figures (9)

Fig. 1
Fig. 1 nSWAT analysis and fabrication. a) A schematic of the device layout. The electric field vectors at different locations along the waveguide are indicated. Input laser beam of power P = B2 is divided at the 50/50 splitter into two counter propagating waves of powers P1 = E2 and P2 = F2 and, each of which is half of P in the case of an ideal splitter. Wave of power P2 passes through a region of the waveguide modulated by a microheater (brown) and acquires an additional phase (θ) before reaching the old trapping regions. Waves of powers P1 and P2 then re-enter the splitter and emerge as waves of powers P' = B'2 at the input arm and O2 at the output. b) An optical microscope image of the fabricated device (false colored for clarity). The device is implemented with silicon nitride waveguides over a 3.8 μm thermal oxide layer on top of a silicon wafer, with waveguide and microheater fabrication details given in [22]. The old and new trapping regions are laid side by side for direct comparison.
Fig. 2
Fig. 2 Standing wave quality in the old and new trapping regions. a) Standing wave ratio as a function of t. b) Power ratio of forward to backward propagating beams as a function of t. c) Scattering power normalized to that at t = 1 as a function of t.
Fig. 3
Fig. 3 Maximum trapping force as a function of bead diameter. Simulations were performed using 1 W total local laser power (0.5 W power from each direction). Parameters used are: 250 nm × 550 nm silicon nitride waveguide, polystyrene beads, and 1064 nm laser, bead to waveguide surface-to-surface distance 5 nm, with other configurations detailed in [22].
Fig. 4
Fig. 4 Direct comparison of the power in a waveguide in the old and new trapping regions. a) Scanning electron micrograph of the waveguides in the two trapping regions. The waveguides are 550 nm in width and 250 nm in height. b) Image of scattered light from the two waveguides. The brightness of the scattered light from each waveguide is proportional to the power transmitted through the waveguide. c) Intensity plot of the waveguides. The intensity plot is generated by taking the image in (b), integrating intensity values along the z axis, and plotting the integrated intensity value along the x direction. The solid red curve is a fit to the sum of two Gaussian functions, corresponding to the scattered light at the old and new trapping region respectively. The cumulative intensity of the scattered light at each region is determined as the area under the corresponding Gaussian. d) Relation in the cumulative intensities of the scattered light in the waveguides in the new and old trapping regions. Measurements were made over a range of input laser power (40 to 160 mW). The resulting plot gives a linear fit of a slope 2.006 ± 0.016.
Fig. 5
Fig. 5 Displacement of a trapped bead versus the square of the voltage applied to the microheater at both new and old trapping regions. The linear fit has a slope of −10.37 ± 0.03 nm/V2 for the old trapping region, and 10.39 ± 0.02 nm/V2 for the new trapping region. Exactly one trapping period is shown for both the new and old trapping regions.
Fig. 6
Fig. 6 Trap reposition resolution of at the new and old trapping regions. The positions of trapped beads at both trapping regions were measured as the traps were repositioned in a square-wave fashion by the microheater, in either 10 nm (a) or 2.5 nm (b) steps. The black and red curves on the bottom plots show the measured bead displacement at the old and new trapping regions, respectively, with solid curves as the fitted bead displacement.
Fig. 7
Fig. 7 Trap stiffness versus laser power in the waveguides. a) An example of trap stiffness measurement using the variance method. A 380-nm polystyrene bead was trapped on the nSWAT in the old trapping region under an estimated 7.2 mW laser power (sum of the powers in both directions) at the trapping region. Local laser power intensities are estimated based on measured loss in each section of the optical path, as detailed in [22]. The bead position distribution (grey) was fit to a Gaussian function (black) to determine the variance of the bead position: σ2 = 516 nm2, yielding trap stiffness: kz = 0.0079 pN/nm. b) Trap stiffness measurement using the power spectrum method. For the same set of data as in a), the power spectrum of the bead position was fit to the Lorentzian function (black) (see text). The fit yielded fc = 103 Hz, and γ = 9.6 × 10−6 pN/(nm•Hz), giving the trap stiffness, kz = 0.0062 pN/nm. c) Trap stiffness versus laser power. At each laser power in the old trapping region, the trap stiffness was determined from both the variance method (open square) and the power spectrum method (open circle) at both the old and new trapping regions. Fitting these results to lines, solid and dashed respectively, gives stiffness 1.04 pN/nm per watt for the variance method and 0.95 pN/nm per watt for the power spectrum method at the old design, and 2.19 pN/nm per watt for the variance method and 2.07 pN/nm per watt for the power spectrum method at the new design. The error bars are standard errors of the mean.
Fig. 8
Fig. 8 Bead velocity along the waveguide versus trap velocity. Two trapped beads, one in the old trapping region and another in the new trapping region, were monitored as the traps were moved at a constant speed. Because the two beads in the two regions moved in opposite directions under microheater modulation, the positive trap speed data at the old trapping region were acquired simultaneously with the negative trap speed data at the new trapping region. Measurements were performed at 7.2 mW laser power in the old trapping region. The red squares are measured bead velocity in the new trapping region, while the black dots are measured bead velocity in the old trapping region. For each region, data in each direction were fitted by a piecewise function based on the theoretical predications [22]. For the new (old) trapping region, the critical velocity from the fit was 69.16 μm/s (38.02 μm/s) in the positive direction, 68.22 μm/s (38.45 μm/s) in the negative direction, averaging to be 68.69 μm/s (38.24 μm/s). Therefore the maximum trapping force (Fmax = γvcritical) was found to be 0.625 pN and 0.347 pN for the new and old region, respectively.
Fig. 9
Fig. 9 Escape power measurement. a) Displacements of trapped beads in the new and old regions as laser power is steadily decreased. Measurements were formed using a device with a gap distance of 620 nm. Different colors correspond to different trapped beads in a given region. From these traces, we determined that the escape power ratio of the old to the new region was 2.30 ± 0.09 for this device. b) Bead escape power ratio of the old to the new trapping region versus edge-to-edge splitter gap distance. At a 560 nm gap distance, the splitter is closest to 50/50 splitting ( t=k=1/ 2 ).

Equations (12)

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

[ E F ]=[ t ik ik t ][ B 0 ]=[ t ik ]B, t 2 + k 2 =1,
[ B O ]=[ t ik ik t ][ F e iθ E e iθ ]=[ t ik ik t ][ ik t ] e iθ =[ 2ikt t 2 k 2 ]B e iθ ,
B e iβz + B e +iβz =( 1+2ikt e iθ+2iβz )B e iβz ,
E e iβz + F e iθ+iβz =( t+ik e iθ+2iβz )B e iβz ,
I new =[1+4 k 2 t 2 4ktsin(2βz+θ)] B 2 ,
I old =[ t 2 + k 2 2ktsin(2βz+θ)] B 2 ,
SWR new = ( 1+2kt ) 2 ( 12kt ) 2 = ( t+k ) 4 ( tk ) 4 ,( since t 2 + k 2 =1 ),
SWR old = ( t+k ) 2 ( tk ) 2 ,
R new = | B | 2 / | B | 2 =4 k 2 t 2 ,
R old = | F | 2 / | E | 2 = k 2 / t 2 ,
S new = 1 2 Re( E × H * ) | B | 2 | B | 2 =( 14 k 2 t 2 ) B 2
S old = 1 2 Re( E × H * ) | E | 2 | F | 2 =( t 2 k 2 ) B 2

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