Abstract

In this study, we present a method designed to generate dynamic holograms in holographic optical tweezers. The approach combines our random mask encoding method with iterative high-efficiency algorithms. This hybrid method can be used to dynamically modify precalculated holograms, giving them new functionalities—temporarily or permanently—with a low computational cost. This allows the easy addition or removal of a single trap or the independent control of groups of traps for manipulating a variety of rigid structures in real time.

© 2011 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  13. R. Holmlin, M. Schiavoni, C. Chen, S. Smith, M. Prentiss, and G. Whitesides, “Light-driven microfabrication: assembly of multicomponent, three-dimensional structures by using optical tweezers,” Angew. Chem., Int. Ed. Engl. 39, 3503–3506(2000).
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    [CrossRef]
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    [CrossRef]
  22. W. Mu, G. Wang, L. Luan, G. C. Spalding, and J. B. Ketterson, “Dynamic control of defects in a two-dimensional optically assisted assembly,” New J. Phys. 8, 70/1–70/7 (2006).
    [CrossRef]
  23. E. Pleguezuelos, A. Carnicer, J. Andilla, E. Martín-Badosa, and M. Montes-Usategui, “Holotrap: interactive hologram design for multiple dynamic optical trapping,” Comput. Phys. Commun. 176, 701–709 (2007).
    [CrossRef]
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    [CrossRef] [PubMed]

2010 (1)

S. Bianchi and R. Di Leonardo, “Real-time optical micro-manipulation using optimized holograms generated on the GPU,” Comput. Phys. Commun. 181, 1444–1448(2010).
[CrossRef]

2009 (1)

2008 (1)

G. Gibson, D. M. Carberry, G. Whyte, J. Leach, J. Courtial, J. C. Jackson, D. Robert, M. Miles, and M. Padgett, “Holographic assembly workstation for optical manipulation,” J. Opt. A: Pure Appl. Opt. 10, 044009 (2008).
[CrossRef]

2007 (3)

E. Pleguezuelos, A. Carnicer, J. Andilla, E. Martín-Badosa, and M. Montes-Usategui, “Holotrap: interactive hologram design for multiple dynamic optical trapping,” Comput. Phys. Commun. 176, 701–709 (2007).
[CrossRef]

E. Martín-Badosa, M. Montes-Usategui, A. Carnicer, J. Andilla, E. Pleguezuelos, and I. Juvells, “Design strategies for optimizing holographic optical tweezers set-ups,” J. Opt. A: Pure Appl. Opt. 9, S267–S277 (2007).
[CrossRef]

R. Di Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express 15, 1913–1922 (2007).
[CrossRef] [PubMed]

2006 (4)

2005 (5)

2002 (2)

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175(2002).
[CrossRef]

P. Korda, G. C. Spalding, E. R. Dufresne, and D. G. Grier, “Nanofabrication with holographic optical tweezers,” Rev. Sci. Instrum. 73, 1956–1957 (2002).
[CrossRef]

2000 (2)

R. Holmlin, M. Schiavoni, C. Chen, S. Smith, M. Prentiss, and G. Whitesides, “Light-driven microfabrication: assembly of multicomponent, three-dimensional structures by using optical tweezers,” Angew. Chem., Int. Ed. Engl. 39, 3503–3506(2000).
[CrossRef]

J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computer-generated holograms,” Opt. Commun. 185, 77–82 (2000).
[CrossRef]

1994 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Agarwal, R.

Andilla, J.

E. Martín-Badosa, M. Montes-Usategui, A. Carnicer, J. Andilla, E. Pleguezuelos, and I. Juvells, “Design strategies for optimizing holographic optical tweezers set-ups,” J. Opt. A: Pure Appl. Opt. 9, S267–S277 (2007).
[CrossRef]

E. Pleguezuelos, A. Carnicer, J. Andilla, E. Martín-Badosa, and M. Montes-Usategui, “Holotrap: interactive hologram design for multiple dynamic optical trapping,” Comput. Phys. Commun. 176, 701–709 (2007).
[CrossRef]

M. Montes-Usategui, E. Pleguezuelos, J. Andilla, and E. Martín-Badosa, “Fast generation of holographic optical tweezers by random mask encoding of Fourier components,” Opt. Express 14, 2101–2107 (2006).
[CrossRef] [PubMed]

Bianchi, S.

S. Bianchi and R. Di Leonardo, “Real-time optical micro-manipulation using optimized holograms generated on the GPU,” Comput. Phys. Commun. 181, 1444–1448(2010).
[CrossRef]

Carberry, D. M.

J. A. Grieve, A. Ulcinas, S. Subramanian, G. M. Gibson, M. J. Padgett, D. M. Carberry, and M. J. Miles, “Hands-on with optical tweezers: a multitouch interface for holographic optical trapping,” Opt. Express 17, 3595–3602(2009).
[CrossRef] [PubMed]

G. Gibson, D. M. Carberry, G. Whyte, J. Leach, J. Courtial, J. C. Jackson, D. Robert, M. Miles, and M. Padgett, “Holographic assembly workstation for optical manipulation,” J. Opt. A: Pure Appl. Opt. 10, 044009 (2008).
[CrossRef]

Carnicer, A.

E. Martín-Badosa, M. Montes-Usategui, A. Carnicer, J. Andilla, E. Pleguezuelos, and I. Juvells, “Design strategies for optimizing holographic optical tweezers set-ups,” J. Opt. A: Pure Appl. Opt. 9, S267–S277 (2007).
[CrossRef]

E. Pleguezuelos, A. Carnicer, J. Andilla, E. Martín-Badosa, and M. Montes-Usategui, “Holotrap: interactive hologram design for multiple dynamic optical trapping,” Comput. Phys. Commun. 176, 701–709 (2007).
[CrossRef]

Castelino, K.

K. Castelino, S. Satyanarayana, and M. Sitti, “Manufacturing of two and three-dimensional micro/nanostructures by integrating optical tweezers with chemical assembly,” Robotica 23, 435–439 (2005).
[CrossRef]

Chapin, S. C.

Chen, C.

R. Holmlin, M. Schiavoni, C. Chen, S. Smith, M. Prentiss, and G. Whitesides, “Light-driven microfabrication: assembly of multicomponent, three-dimensional structures by using optical tweezers,” Angew. Chem., Int. Ed. Engl. 39, 3503–3506(2000).
[CrossRef]

Cottrell, D. M.

Courtial, J.

G. Gibson, D. M. Carberry, G. Whyte, J. Leach, J. Courtial, J. C. Jackson, D. Robert, M. Miles, and M. Padgett, “Holographic assembly workstation for optical manipulation,” J. Opt. A: Pure Appl. Opt. 10, 044009 (2008).
[CrossRef]

G. C. Spalding, J. Courtial, and R. Di Leonardo, “Holographic optical tweezers,” in Structured Light and Its Applications, D.L.Andrews, ed. (Academic Press, 2008), pp. 139–168.
[CrossRef]

Creely, C.

Curtis, J. E.

J. E. Curtis, C. H. J. Schmitz, and J. P. Spatz, “Symmetry dependence of holograms for optical trapping,” Opt. Lett. 30, 2086–2088 (2005).
[CrossRef] [PubMed]

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175(2002).
[CrossRef]

Davis, J. A.

Di Leonardo, R.

S. Bianchi and R. Di Leonardo, “Real-time optical micro-manipulation using optimized holograms generated on the GPU,” Comput. Phys. Commun. 181, 1444–1448(2010).
[CrossRef]

R. Di Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express 15, 1913–1922 (2007).
[CrossRef] [PubMed]

G. C. Spalding, J. Courtial, and R. Di Leonardo, “Holographic optical tweezers,” in Structured Light and Its Applications, D.L.Andrews, ed. (Academic Press, 2008), pp. 139–168.
[CrossRef]

Dufresne, E. R.

S. C. Chapin, V. Germain, and E. R. Dufresne, “Automated trapping, assembly, and sorting with holographic optical tweezers,” Opt. Express 14, 13095–13100 (2006).
[CrossRef] [PubMed]

P. Korda, G. C. Spalding, E. R. Dufresne, and D. G. Grier, “Nanofabrication with holographic optical tweezers,” Rev. Sci. Instrum. 73, 1956–1957 (2002).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Germain, V.

Gibson, G.

G. Gibson, D. M. Carberry, G. Whyte, J. Leach, J. Courtial, J. C. Jackson, D. Robert, M. Miles, and M. Padgett, “Holographic assembly workstation for optical manipulation,” J. Opt. A: Pure Appl. Opt. 10, 044009 (2008).
[CrossRef]

Gibson, G. M.

Grier, D.

Grier, D. G.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175(2002).
[CrossRef]

P. Korda, G. C. Spalding, E. R. Dufresne, and D. G. Grier, “Nanofabrication with holographic optical tweezers,” Rev. Sci. Instrum. 73, 1956–1957 (2002).
[CrossRef]

Grieve, J. A.

Haist, T.

Holmlin, R.

R. Holmlin, M. Schiavoni, C. Chen, S. Smith, M. Prentiss, and G. Whitesides, “Light-driven microfabrication: assembly of multicomponent, three-dimensional structures by using optical tweezers,” Angew. Chem., Int. Ed. Engl. 39, 3503–3506(2000).
[CrossRef]

Ianni, F.

Jackson, J. C.

G. Gibson, D. M. Carberry, G. Whyte, J. Leach, J. Courtial, J. C. Jackson, D. Robert, M. Miles, and M. Padgett, “Holographic assembly workstation for optical manipulation,” J. Opt. A: Pure Appl. Opt. 10, 044009 (2008).
[CrossRef]

Juvells, I.

E. Martín-Badosa, M. Montes-Usategui, A. Carnicer, J. Andilla, E. Pleguezuelos, and I. Juvells, “Design strategies for optimizing holographic optical tweezers set-ups,” J. Opt. A: Pure Appl. Opt. 9, S267–S277 (2007).
[CrossRef]

Ketterson, J. B.

W. Mu, G. Wang, L. Luan, G. C. Spalding, and J. B. Ketterson, “Dynamic control of defects in a two-dimensional optically assisted assembly,” New J. Phys. 8, 70/1–70/7 (2006).
[CrossRef]

Kohler, C.

Korda, P.

P. Korda, G. C. Spalding, E. R. Dufresne, and D. G. Grier, “Nanofabrication with holographic optical tweezers,” Rev. Sci. Instrum. 73, 1956–1957 (2002).
[CrossRef]

Koss, B. A.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175(2002).
[CrossRef]

Ladavac, K.

Leach, J.

G. Gibson, D. M. Carberry, G. Whyte, J. Leach, J. Courtial, J. C. Jackson, D. Robert, M. Miles, and M. Padgett, “Holographic assembly workstation for optical manipulation,” J. Opt. A: Pure Appl. Opt. 10, 044009 (2008).
[CrossRef]

Lee, S.-H.

Lieber, C.

Liesener, J.

J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computer-generated holograms,” Opt. Commun. 185, 77–82 (2000).
[CrossRef]

Luan, L.

W. Mu, G. Wang, L. Luan, G. C. Spalding, and J. B. Ketterson, “Dynamic control of defects in a two-dimensional optically assisted assembly,” New J. Phys. 8, 70/1–70/7 (2006).
[CrossRef]

Martín-Badosa, E.

E. Martín-Badosa, M. Montes-Usategui, A. Carnicer, J. Andilla, E. Pleguezuelos, and I. Juvells, “Design strategies for optimizing holographic optical tweezers set-ups,” J. Opt. A: Pure Appl. Opt. 9, S267–S277 (2007).
[CrossRef]

E. Pleguezuelos, A. Carnicer, J. Andilla, E. Martín-Badosa, and M. Montes-Usategui, “Holotrap: interactive hologram design for multiple dynamic optical trapping,” Comput. Phys. Commun. 176, 701–709 (2007).
[CrossRef]

M. Montes-Usategui, E. Pleguezuelos, J. Andilla, and E. Martín-Badosa, “Fast generation of holographic optical tweezers by random mask encoding of Fourier components,” Opt. Express 14, 2101–2107 (2006).
[CrossRef] [PubMed]

Miles, M.

G. Gibson, D. M. Carberry, G. Whyte, J. Leach, J. Courtial, J. C. Jackson, D. Robert, M. Miles, and M. Padgett, “Holographic assembly workstation for optical manipulation,” J. Opt. A: Pure Appl. Opt. 10, 044009 (2008).
[CrossRef]

Miles, M. J.

Montes-Usategui, M.

E. Martín-Badosa, M. Montes-Usategui, A. Carnicer, J. Andilla, E. Pleguezuelos, and I. Juvells, “Design strategies for optimizing holographic optical tweezers set-ups,” J. Opt. A: Pure Appl. Opt. 9, S267–S277 (2007).
[CrossRef]

E. Pleguezuelos, A. Carnicer, J. Andilla, E. Martín-Badosa, and M. Montes-Usategui, “Holotrap: interactive hologram design for multiple dynamic optical trapping,” Comput. Phys. Commun. 176, 701–709 (2007).
[CrossRef]

M. Montes-Usategui, E. Pleguezuelos, J. Andilla, and E. Martín-Badosa, “Fast generation of holographic optical tweezers by random mask encoding of Fourier components,” Opt. Express 14, 2101–2107 (2006).
[CrossRef] [PubMed]

Mu, W.

W. Mu, G. Wang, L. Luan, G. C. Spalding, and J. B. Ketterson, “Dynamic control of defects in a two-dimensional optically assisted assembly,” New J. Phys. 8, 70/1–70/7 (2006).
[CrossRef]

Osten, W.

Padgett, M.

G. Gibson, D. M. Carberry, G. Whyte, J. Leach, J. Courtial, J. C. Jackson, D. Robert, M. Miles, and M. Padgett, “Holographic assembly workstation for optical manipulation,” J. Opt. A: Pure Appl. Opt. 10, 044009 (2008).
[CrossRef]

Padgett, M. J.

Petrov, D.

Pleguezuelos, E.

E. Martín-Badosa, M. Montes-Usategui, A. Carnicer, J. Andilla, E. Pleguezuelos, and I. Juvells, “Design strategies for optimizing holographic optical tweezers set-ups,” J. Opt. A: Pure Appl. Opt. 9, S267–S277 (2007).
[CrossRef]

E. Pleguezuelos, A. Carnicer, J. Andilla, E. Martín-Badosa, and M. Montes-Usategui, “Holotrap: interactive hologram design for multiple dynamic optical trapping,” Comput. Phys. Commun. 176, 701–709 (2007).
[CrossRef]

M. Montes-Usategui, E. Pleguezuelos, J. Andilla, and E. Martín-Badosa, “Fast generation of holographic optical tweezers by random mask encoding of Fourier components,” Opt. Express 14, 2101–2107 (2006).
[CrossRef] [PubMed]

Polin, M.

Prentiss, M.

R. Holmlin, M. Schiavoni, C. Chen, S. Smith, M. Prentiss, and G. Whitesides, “Light-driven microfabrication: assembly of multicomponent, three-dimensional structures by using optical tweezers,” Angew. Chem., Int. Ed. Engl. 39, 3503–3506(2000).
[CrossRef]

Reicherter, M.

Robert, D.

G. Gibson, D. M. Carberry, G. Whyte, J. Leach, J. Courtial, J. C. Jackson, D. Robert, M. Miles, and M. Padgett, “Holographic assembly workstation for optical manipulation,” J. Opt. A: Pure Appl. Opt. 10, 044009 (2008).
[CrossRef]

Roichman, Y.

Ruocco, G.

Satyanarayana, S.

K. Castelino, S. Satyanarayana, and M. Sitti, “Manufacturing of two and three-dimensional micro/nanostructures by integrating optical tweezers with chemical assembly,” Robotica 23, 435–439 (2005).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Schiavoni, M.

R. Holmlin, M. Schiavoni, C. Chen, S. Smith, M. Prentiss, and G. Whitesides, “Light-driven microfabrication: assembly of multicomponent, three-dimensional structures by using optical tweezers,” Angew. Chem., Int. Ed. Engl. 39, 3503–3506(2000).
[CrossRef]

Schmitz, C. H. J.

Singh, G.

Sitti, M.

K. Castelino, S. Satyanarayana, and M. Sitti, “Manufacturing of two and three-dimensional micro/nanostructures by integrating optical tweezers with chemical assembly,” Robotica 23, 435–439 (2005).
[CrossRef]

Smith, S.

R. Holmlin, M. Schiavoni, C. Chen, S. Smith, M. Prentiss, and G. Whitesides, “Light-driven microfabrication: assembly of multicomponent, three-dimensional structures by using optical tweezers,” Angew. Chem., Int. Ed. Engl. 39, 3503–3506(2000).
[CrossRef]

Soler, M.

Spalding, G. C.

W. Mu, G. Wang, L. Luan, G. C. Spalding, and J. B. Ketterson, “Dynamic control of defects in a two-dimensional optically assisted assembly,” New J. Phys. 8, 70/1–70/7 (2006).
[CrossRef]

P. Korda, G. C. Spalding, E. R. Dufresne, and D. G. Grier, “Nanofabrication with holographic optical tweezers,” Rev. Sci. Instrum. 73, 1956–1957 (2002).
[CrossRef]

G. C. Spalding, J. Courtial, and R. Di Leonardo, “Holographic optical tweezers,” in Structured Light and Its Applications, D.L.Andrews, ed. (Academic Press, 2008), pp. 139–168.
[CrossRef]

Spatz, J. P.

Subramanian, S.

Tiziani, H.

Tiziani, H. J.

J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computer-generated holograms,” Opt. Commun. 185, 77–82 (2000).
[CrossRef]

Ulcinas, A.

Volpe, G.

Wang, G.

W. Mu, G. Wang, L. Luan, G. C. Spalding, and J. B. Ketterson, “Dynamic control of defects in a two-dimensional optically assisted assembly,” New J. Phys. 8, 70/1–70/7 (2006).
[CrossRef]

Whitesides, G.

R. Holmlin, M. Schiavoni, C. Chen, S. Smith, M. Prentiss, and G. Whitesides, “Light-driven microfabrication: assembly of multicomponent, three-dimensional structures by using optical tweezers,” Angew. Chem., Int. Ed. Engl. 39, 3503–3506(2000).
[CrossRef]

Whyte, G.

G. Gibson, D. M. Carberry, G. Whyte, J. Leach, J. Courtial, J. C. Jackson, D. Robert, M. Miles, and M. Padgett, “Holographic assembly workstation for optical manipulation,” J. Opt. A: Pure Appl. Opt. 10, 044009 (2008).
[CrossRef]

Yu, G.

Zwick, S.

Angew. Chem., Int. Ed. Engl. (1)

R. Holmlin, M. Schiavoni, C. Chen, S. Smith, M. Prentiss, and G. Whitesides, “Light-driven microfabrication: assembly of multicomponent, three-dimensional structures by using optical tweezers,” Angew. Chem., Int. Ed. Engl. 39, 3503–3506(2000).
[CrossRef]

Appl. Opt. (1)

Comput. Phys. Commun. (2)

S. Bianchi and R. Di Leonardo, “Real-time optical micro-manipulation using optimized holograms generated on the GPU,” Comput. Phys. Commun. 181, 1444–1448(2010).
[CrossRef]

E. Pleguezuelos, A. Carnicer, J. Andilla, E. Martín-Badosa, and M. Montes-Usategui, “Holotrap: interactive hologram design for multiple dynamic optical trapping,” Comput. Phys. Commun. 176, 701–709 (2007).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (2)

G. Gibson, D. M. Carberry, G. Whyte, J. Leach, J. Courtial, J. C. Jackson, D. Robert, M. Miles, and M. Padgett, “Holographic assembly workstation for optical manipulation,” J. Opt. A: Pure Appl. Opt. 10, 044009 (2008).
[CrossRef]

E. Martín-Badosa, M. Montes-Usategui, A. Carnicer, J. Andilla, E. Pleguezuelos, and I. Juvells, “Design strategies for optimizing holographic optical tweezers set-ups,” J. Opt. A: Pure Appl. Opt. 9, S267–S277 (2007).
[CrossRef]

New J. Phys. (1)

W. Mu, G. Wang, L. Luan, G. C. Spalding, and J. B. Ketterson, “Dynamic control of defects in a two-dimensional optically assisted assembly,” New J. Phys. 8, 70/1–70/7 (2006).
[CrossRef]

Opt. Commun. (2)

J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computer-generated holograms,” Opt. Commun. 185, 77–82 (2000).
[CrossRef]

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175(2002).
[CrossRef]

Opt. Express (7)

Opt. Lett. (2)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Rev. Sci. Instrum. (1)

P. Korda, G. C. Spalding, E. R. Dufresne, and D. G. Grier, “Nanofabrication with holographic optical tweezers,” Rev. Sci. Instrum. 73, 1956–1957 (2002).
[CrossRef]

Robotica (1)

K. Castelino, S. Satyanarayana, and M. Sitti, “Manufacturing of two and three-dimensional micro/nanostructures by integrating optical tweezers with chemical assembly,” Robotica 23, 435–439 (2005).
[CrossRef]

Other (3)

Arryx: BioRyx 200 User manual, http://www.arryx.com/PDFdocs/BioRyx200manual_2ndEd.pdf.

Computer Integrated Systems for Microscopy and Manipulation, http://cismm.cs.unc.edu/.

G. C. Spalding, J. Courtial, and R. Di Leonardo, “Holographic optical tweezers,” in Structured Light and Its Applications, D.L.Andrews, ed. (Academic Press, 2008), pp. 139–168.
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Supplementary Material (3)

» Media 1: AVI (4766 KB)     
» Media 2: AVI (390 KB)     
» Media 3: AVI (3881 KB)     

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Figures (6)

Fig. 1
Fig. 1

Manipulation of two 4-trap parts. The two GS holograms were multiplexed using the RM encoding method. From (a) to (b) a linear phase has been added to one of the pixel masks in order to displace the squared group of traps while keeping the other piece static (see Media 1 for a rotation demonstration).

Fig. 2
Fig. 2

Video sequence showing the independent manipulation of three blocks of four trapped beads ( diameter = 1 μm ). One GS hologram was computed for each block. The three holograms were multiplexed using the RM encoding method. Independent manipulation of each part is achieved by adding linear phases or rotating the hologram in 2D within each pixel mask (see Media 2).

Fig. 3
Fig. 3

The efficiencies of the holograms used for the experiment in Fig. 4 were evaluated numerically by Fourier transforming each phase hologram to obtain the energy distribution at the sample plane. Efficiency was calculated as the ratio between the amount of energy at trap positions (computed by adding the square of the image values in the trap centers and its immediate neighbor pixels) and the total energy of the image. The spots with error bars show the efficiency of the holograms as a function of the number of multiplexed holograms (N). Error bars are obtained from trap intensity variation in the simulated image of the trap pattern. The solid line shows the fitted power law ( efficiency = a × N ^ b ) in the logarithmic graph, with a = 0.61 ± 0.01 and b = 1.01 ± 0.04 .

Fig. 4
Fig. 4

Experimental performance of the hybrid method for generating a 4 × 3 array of optical traps. Data show the mean experimental transverse trap stiffness k (averaged for the whole array) as a function of the number of multiplexed holograms (N). The values are relative to a reference stiffness k ref = 27 ± 7 pN / μm ), which corresponds to that of traps in the pure GS hologram. Error bars arise from the variability of stiffness in the array, which may come from the small differences in particle size, or possible power fluctuations from trap to trap. The solid line shows the fitting to an exponential function (negative) in the logarithmic graph, resulting in a power law ( relative stiffness = a × N ^ b ) with a = 1.00 ± 0.08 and b = 1.6 ± 0.2 .

Fig. 5
Fig. 5

An array of 41 traps created with a Gerchberg–Saxton hologram was loaded with beads of three different sizes using an auxiliary trap. (a) An auxiliary trap (here marked with a black circle) is used to place the beads in the desired positions. (b) After the pattern is finished the extra trap can be removed (see Media 3).

Fig. 6
Fig. 6

Elimination of a trap from an existing pattern. (a) The trap marked with a yellow circle is superimposed with an auxiliary trap. (b) If the phase and percentage are chosen correctly, the trap is extinguished.

Tables (1)

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Table 1 Component Holograms (GS) for Building a 4 × 3 Array of Optical Traps Using Hybrid Method a

Equations (9)

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H ( u , v ) = l = 1 N m l ( u , v ) · H l ( u , v ) ,
m l ( u , v ) = { 1 ( u , v ) M l 0 ( u , v ) M l .
E tot = j = 1 P k = 1 P ( H ( j , k ) A ) * · H ( j , k ) A = j = 1 P k = 1 P | H ( j , k ) | 2 = P × P = P 2 .
C ( x , y ) = 1 P j = 1 P k = 1 P H ( j , k ) · e i 2 π P ( x · j + y · k ) .
C ( x t , y t ) = 1 P j = 1 P k = 1 P l = 1 N m l ( j , k ) · e i 2 π P ( ( x t x l ) · j + ( y t y l ) · k ) 1 P j = 1 P k = 1 P m t ( j , k ) = n t P ,
E traps = t = 1 N ( n t P ) 2 .
η = E traps E total = 1 P 2 t = 1 N n t 2 P 2 = t = 1 N 1 N 2 = 1 N .
R = K · M · P 2 U = K · M · B ,
k x = k B T σ x 2 , k y = k B T σ y 2 ,

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