Abstract

We introduce a method whereby the electromagnetic field that governs the force on a Rayleigh particle can be tailored such that the resultant force field conforms to a desired geometry. The electromagnetic field is expanded as a set of vector spherical wavefunctions (VSWFs) that describe the field over all space. Given the incident field, the resultant force on a given Rayleigh particle can be calculated throughout a volume of interest. We use an evolutionary algorithm (EA) to search the space of coefficients governing the VSWFs for those that produce the desired force field. We demonstrate how Maxwell’s equations will support an “optical tunnel” that guides particles to a trap location while at the same time preventing particles outside the tunnel from approaching the trap. This result is of interest because the field is impressed throughout the domain; that is to say, once the field is generated, no additional control is required to guide the particles.

© 2011 OSA

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2010

D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light forces the pace: optical manipulation for biophotonics,” J. Biomed. Opt. 15, 041503 (2010).
[CrossRef] [PubMed]

T. Čižmár, L. C. Dávila Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B 43, 102001 (2010).
[CrossRef]

2009

J. B. Wills, J. R. Butler, J. Palmer, and J. P. Reid, “Using optical landscapes to control, direct, and isolate aerosol particles,” Phys. Chem. Chem. Phys. 11, 8015–8020 (2009).
[CrossRef] [PubMed]

S. H. Simpson and S. Hanna, “Rotation of absorbing spheres in Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 26, 173–183 (2009).
[CrossRef]

2008

B. Sun, Y. Roichman, and D. G. Grier, “Theory of holographic optical trapping,” Opt. Express 16, 15765–15776 (2008).
[CrossRef] [PubMed]

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

M. T. Valentine, N. R. Guydosh, B. Gutiérrez-Medina, A. N. Fehr, J. O. Andreasson, and S. M. Block, “Precision steering of an optical trap by electro-optic deflection,” Opt. Commun. 33, 599–601 (2008).

2007

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007).
[CrossRef]

2005

G. Whyte and J. Courtial, “Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg–Saxton algorithm,” New J. Phys. 7, 117 (2005).
[CrossRef]

P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Four-dimensional optical manipulation of colloidal particles,” App. Phys. Lett. 86, 074103 (2005).
[CrossRef]

T. Čižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” App. Phys. Lett. 86, 174101 (2005).
[CrossRef]

V. Garcés-Chávez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticies on a surface,” App. Phys. Lett. 86, 031106 (2005).
[CrossRef]

O. Moine and B. Stout, “Optical force calculations in arbitrary beams by use of the vector addition theorem,” J. Opt. Soc. Am. B 22, 1620–1631 (2005).
[CrossRef]

2004

2003

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

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. Shabtay, “Three-dimensional beam forming and Ewald’s surfaces,” Opt. Commun. 226, 33–37 (2003).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1019–1029 (2003).
[CrossRef]

2002

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

V. Garcés-Chávez, D. McGloin, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[CrossRef] [PubMed]

2001

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[CrossRef]

2000

P. C. Morgensen and J. Glückstad, “Dynamic array generation and pattern formation for optical tweezers,” Opt. Commun. 175, 75–81 (2000).
[CrossRef]

C. Mio, T. Gong, A. Terray, and D. W. M. Marr, “Design of a scanning laser optical trap for multiparticle manipulation,” Rev. Sci. Instrum. 71, 2196–2200 (2000).
[CrossRef]

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

J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. 25, 191–193 (2000).
[CrossRef]

1999

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, 8825–8831 (1999).
[CrossRef]

M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999).
[CrossRef]

1998

E. R. Dufresne and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optics,” Rev. Sci. Instrum. 69, 1974–1977 (1998).
[CrossRef]

1997

A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. 133, 7–10 (1997).
[CrossRef]

R. Storn and R. Price, “Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. 11, 341–359 (1997).
[CrossRef]

1996

R. Piestun, B. Spektor, and J. Shamir, “Unconventional light distributions in three-dimensional domains,” J. Mod. Opt. 43, 1495–1507 (1996).
[CrossRef]

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

R. Piestun, B. Spektor, and J. Shamir, “Wave fields in three dimensions: analysis and synthesis,” J. Opt. Soc. Am. A 13, 1837–1848 (1996).
[CrossRef]

K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Sel. Top. Quantum Electron. 2, 1066–1075 (1996).
[CrossRef]

K. T. Gahagan and G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996).
[CrossRef] [PubMed]

1995

1994

1992

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, “Optical trapping of a metal-particle and a water droplet by a scanning laser-beam,” App. Phys. Lett. 60, 807–809 (1992).
[CrossRef]

1987

1986

1979

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

1972

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

Allebach, J. P.

Andreasson, J. O.

M. T. Valentine, N. R. Guydosh, B. Gutiérrez-Medina, A. N. Fehr, J. O. Andreasson, and S. M. Block, “Precision steering of an optical trap by electro-optic deflection,” Opt. Commun. 33, 599–601 (2008).

Andrews, D. L.

T. Čižmár, L. C. Dávila Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B 43, 102001 (2010).
[CrossRef]

D. L. Andrews, Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Elsevier, 2008).
[PubMed]

Arlt, J.

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[CrossRef]

J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. 25, 191–193 (2000).
[CrossRef]

Asakura, T.

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

Ashkin, A.

Berns, M. W.

A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. 133, 7–10 (1997).
[CrossRef]

Bjorkholm, J. E.

Block, S. M.

M. T. Valentine, N. R. Guydosh, B. Gutiérrez-Medina, A. N. Fehr, J. O. Andreasson, and S. M. Block, “Precision steering of an optical trap by electro-optic deflection,” Opt. Commun. 33, 599–601 (2008).

K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Sel. Top. Quantum Electron. 2, 1066–1075 (1996).
[CrossRef]

Branczyk, A. M.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007).
[CrossRef]

Butler, J. R.

J. B. Wills, J. R. Butler, J. Palmer, and J. P. Reid, “Using optical landscapes to control, direct, and isolate aerosol particles,” Phys. Chem. Chem. Phys. 11, 8015–8020 (2009).
[CrossRef] [PubMed]

Chiou, A. E.

A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. 133, 7–10 (1997).
[CrossRef]

Choi, C. H.

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, 8825–8831 (1999).
[CrossRef]

Chu, S.

Cižmár, T.

T. Čižmár, L. C. Dávila Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B 43, 102001 (2010).
[CrossRef]

T. Čižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” App. Phys. Lett. 86, 174101 (2005).
[CrossRef]

Cooper, J.

Courtial, J.

G. Whyte and J. Courtial, “Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg–Saxton algorithm,” New J. Phys. 7, 117 (2005).
[CrossRef]

J. Leach, G. Sinclair, P. Jordan, J. Courtial, M. Padgett, J. Cooper, and Z. Laczik, “3D manipulation of particles into crystal structures using holographic optical tweezers,” Opt. Express 12, 220–226 (2004).
[CrossRef] [PubMed]

G. C. Spalding, J. Courtial, and R. Di Leonardo “Holographic Optical Tweezers,” in Structured Light and Its Applications, D. L. Andrews, ed. (Elsevier, 2008), Chap. 6.
[CrossRef]

Curtis, J.

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

Daria, V. R.

P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Four-dimensional optical manipulation of colloidal particles,” App. Phys. Lett. 86, 074103 (2005).
[CrossRef]

Dávila Romero, L. C.

T. Čižmár, L. C. Dávila Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B 43, 102001 (2010).
[CrossRef]

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Dholakia, K.

D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light forces the pace: optical manipulation for biophotonics,” J. Biomed. Opt. 15, 041503 (2010).
[CrossRef] [PubMed]

T. Čižmár, L. C. Dávila Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B 43, 102001 (2010).
[CrossRef]

T. Čižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” App. Phys. Lett. 86, 174101 (2005).
[CrossRef]

V. Garcés-Chávez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticies on a surface,” App. Phys. Lett. 86, 031106 (2005).
[CrossRef]

V. Garcés-Chávez, D. McGloin, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[CrossRef] [PubMed]

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[CrossRef]

Di Leonardo, R.

G. C. Spalding, J. Courtial, and R. Di Leonardo “Holographic Optical Tweezers,” in Structured Light and Its Applications, D. L. Andrews, ed. (Elsevier, 2008), Chap. 6.
[CrossRef]

Dufresne, E. R.

E. R. Dufresne and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optics,” Rev. Sci. Instrum. 69, 1974–1977 (1998).
[CrossRef]

Dziedzic, J. M.

Fehr, A. N.

M. T. Valentine, N. R. Guydosh, B. Gutiérrez-Medina, A. N. Fehr, J. O. Andreasson, and S. M. Block, “Precision steering of an optical trap by electro-optic deflection,” Opt. Commun. 33, 599–601 (2008).

Gahagan, K. T.

Garcés-Chávez, V.

V. Garcés-Chávez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticies on a surface,” App. Phys. Lett. 86, 031106 (2005).
[CrossRef]

T. Čižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” App. Phys. Lett. 86, 174101 (2005).
[CrossRef]

V. Garcés-Chávez, D. McGloin, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[CrossRef] [PubMed]

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[CrossRef]

Gerchberg, R. W.

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

Glückstad, J.

P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Four-dimensional optical manipulation of colloidal particles,” App. Phys. Lett. 86, 074103 (2005).
[CrossRef]

P. C. Morgensen and J. Glückstad, “Dynamic array generation and pattern formation for optical tweezers,” Opt. Commun. 175, 75–81 (2000).
[CrossRef]

Gong, T.

C. Mio, T. Gong, A. Terray, and D. W. M. Marr, “Design of a scanning laser optical trap for multiparticle manipulation,” Rev. Sci. Instrum. 71, 2196–2200 (2000).
[CrossRef]

Gordon, M. S.

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, 8825–8831 (1999).
[CrossRef]

Gouesebet, G.

Gréhan, G.

Grier, D. G.

B. Sun, Y. Roichman, and D. G. Grier, “Theory of holographic optical trapping,” Opt. Express 16, 15765–15776 (2008).
[CrossRef] [PubMed]

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

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

E. R. Dufresne and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optics,” Rev. Sci. Instrum. 69, 1974–1977 (1998).
[CrossRef]

Gross, S. P.

K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Sel. Top. Quantum Electron. 2, 1066–1075 (1996).
[CrossRef]

Gunn-Moore, F.

D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light forces the pace: optical manipulation for biophotonics,” J. Biomed. Opt. 15, 041503 (2010).
[CrossRef] [PubMed]

Gutiérrez-Medina, B.

M. T. Valentine, N. R. Guydosh, B. Gutiérrez-Medina, A. N. Fehr, J. O. Andreasson, and S. M. Block, “Precision steering of an optical trap by electro-optic deflection,” Opt. Commun. 33, 599–601 (2008).

Guydosh, N. R.

M. T. Valentine, N. R. Guydosh, B. Gutiérrez-Medina, A. N. Fehr, J. O. Andreasson, and S. M. Block, “Precision steering of an optical trap by electro-optic deflection,” Opt. Commun. 33, 599–601 (2008).

Haist, T.

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

M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999).
[CrossRef]

Hanna, S.

Harada, Y.

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

Heckenberg, N. R.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007).
[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]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1019–1029 (2003).
[CrossRef]

Hong, J.

A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. 133, 7–10 (1997).
[CrossRef]

Ivanic, J.

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, 8825–8831 (1999).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

Jonáš, A.

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

Jordan, P.

Kitamura, N.

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, “Optical trapping of a metal-particle and a water droplet by a scanning laser-beam,” App. Phys. Lett. 60, 807–809 (1992).
[CrossRef]

Knöner, G.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007).
[CrossRef]

Kong, J. A.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Koshioka, M.

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, “Optical trapping of a metal-particle and a water droplet by a scanning laser-beam,” App. Phys. Lett. 60, 807–809 (1992).
[CrossRef]

Koss, B. A.

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

Laczik, Z.

Leach, J.

Liesner, J.

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

Locke, J. A.

Loke, V. L. Y.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007).
[CrossRef]

Marr, D. W. M.

C. Mio, T. Gong, A. Terray, and D. W. M. Marr, “Design of a scanning laser optical trap for multiparticle manipulation,” Rev. Sci. Instrum. 71, 2196–2200 (2000).
[CrossRef]

Masuhara, H.

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, “Optical trapping of a metal-particle and a water droplet by a scanning laser-beam,” App. Phys. Lett. 60, 807–809 (1992).
[CrossRef]

McGloin, D.

V. Garcés-Chávez, D. McGloin, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[CrossRef] [PubMed]

Mio, C.

C. Mio, T. Gong, A. Terray, and D. W. M. Marr, “Design of a scanning laser optical trap for multiparticle manipulation,” Rev. Sci. Instrum. 71, 2196–2200 (2000).
[CrossRef]

Misawa, H.

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, “Optical trapping of a metal-particle and a water droplet by a scanning laser-beam,” App. Phys. Lett. 60, 807–809 (1992).
[CrossRef]

Moine, O.

Morgensen, P. C.

P. C. Morgensen and J. Glückstad, “Dynamic array generation and pattern formation for optical tweezers,” Opt. Commun. 175, 75–81 (2000).
[CrossRef]

Nieminen, T. A.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007).
[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]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1019–1029 (2003).
[CrossRef]

Padgett, M.

Padgett, M. J.

Palmer, J.

J. B. Wills, J. R. Butler, J. Palmer, and J. P. Reid, “Using optical landscapes to control, direct, and isolate aerosol particles,” Phys. Chem. Chem. Phys. 11, 8015–8020 (2009).
[CrossRef] [PubMed]

Piestun, R.

Price, R.

R. Storn and R. Price, “Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. 11, 341–359 (1997).
[CrossRef]

Reicherter, M.

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

M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999).
[CrossRef]

Reid, J. P.

J. B. Wills, J. R. Butler, J. Palmer, and J. P. Reid, “Using optical landscapes to control, direct, and isolate aerosol particles,” Phys. Chem. Chem. Phys. 11, 8015–8020 (2009).
[CrossRef] [PubMed]

Rodrigo, P. J.

P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Four-dimensional optical manipulation of colloidal particles,” App. Phys. Lett. 86, 074103 (2005).
[CrossRef]

Roichman, Y.

Rubinsztein-Dunlop, H.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007).
[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]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1019–1029 (2003).
[CrossRef]

Ruedenberg, K.

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, 8825–8831 (1999).
[CrossRef]

Sasaki, K.

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, “Optical trapping of a metal-particle and a water droplet by a scanning laser-beam,” App. Phys. Lett. 60, 807–809 (1992).
[CrossRef]

Saxton, W. O.

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

Seldowitz, M. A.

Shabtay, G.

G. Shabtay, “Three-dimensional beam forming and Ewald’s surfaces,” Opt. Commun. 226, 33–37 (2003).
[CrossRef]

Shamir, J.

Shin, R. T.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Sibbett, W.

V. Garcés-Chávez, D. McGloin, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[CrossRef] [PubMed]

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[CrossRef]

Simpson, S. H.

Sinclair, G.

Sonek, G. J.

A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. 133, 7–10 (1997).
[CrossRef]

Spalding, G. C.

V. Garcés-Chávez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticies on a surface,” App. Phys. Lett. 86, 031106 (2005).
[CrossRef]

G. C. Spalding, J. Courtial, and R. Di Leonardo “Holographic Optical Tweezers,” in Structured Light and Its Applications, D. L. Andrews, ed. (Elsevier, 2008), Chap. 6.
[CrossRef]

Spektor, B.

R. Piestun, B. Spektor, and J. Shamir, “Unconventional light distributions in three-dimensional domains,” J. Mod. Opt. 43, 1495–1507 (1996).
[CrossRef]

R. Piestun, B. Spektor, and J. Shamir, “Wave fields in three dimensions: analysis and synthesis,” J. Opt. Soc. Am. A 13, 1837–1848 (1996).
[CrossRef]

Stevenson, D. J.

D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light forces the pace: optical manipulation for biophotonics,” J. Biomed. Opt. 15, 041503 (2010).
[CrossRef] [PubMed]

Stilgoe, A. B.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007).
[CrossRef]

Storn, R.

R. Storn and R. Price, “Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. 11, 341–359 (1997).
[CrossRef]

Stout, B.

Sun, B.

Swartzlander, G. A.

Sweeney, D. W.

Terray, A.

C. Mio, T. Gong, A. Terray, and D. W. M. Marr, “Design of a scanning laser optical trap for multiparticle manipulation,” Rev. Sci. Instrum. 71, 2196–2200 (2000).
[CrossRef]

Tiziani, H. J.

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

M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999).
[CrossRef]

Tsang, L.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Valentine, M. T.

M. T. Valentine, N. R. Guydosh, B. Gutiérrez-Medina, A. N. Fehr, J. O. Andreasson, and S. M. Block, “Precision steering of an optical trap by electro-optic deflection,” Opt. Commun. 33, 599–601 (2008).

Videen, G.

G. Videen, “Light Scattering from a Sphere Near a Plane Surface,” in Light Scattering from Microstructures, Lecture Notes in Physics Volume 534, F. Moreno and F. González, eds. (Springer, 2000), Chapter 5.
[CrossRef]

Visscher, K.

K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Sel. Top. Quantum Electron. 2, 1066–1075 (1996).
[CrossRef]

Wagemann, E. U.

Wang, W.

A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. 133, 7–10 (1997).
[CrossRef]

Whyte, G.

G. Whyte and J. Courtial, “Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg–Saxton algorithm,” New J. Phys. 7, 117 (2005).
[CrossRef]

Wills, J. B.

J. B. Wills, J. R. Butler, J. Palmer, and J. P. Reid, “Using optical landscapes to control, direct, and isolate aerosol particles,” Phys. Chem. Chem. Phys. 11, 8015–8020 (2009).
[CrossRef] [PubMed]

Zemánek, P.

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

T. Čižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” App. Phys. Lett. 86, 174101 (2005).
[CrossRef]

App. Phys. Lett.

P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Four-dimensional optical manipulation of colloidal particles,” App. Phys. Lett. 86, 074103 (2005).
[CrossRef]

T. Čižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” App. Phys. Lett. 86, 174101 (2005).
[CrossRef]

V. Garcés-Chávez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticies on a surface,” App. Phys. Lett. 86, 031106 (2005).
[CrossRef]

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, “Optical trapping of a metal-particle and a water droplet by a scanning laser-beam,” App. Phys. Lett. 60, 807–809 (1992).
[CrossRef]

Appl. Opt.

Electrophoresis

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

IEEE J. Sel. Top. Quantum Electron.

K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Sel. Top. Quantum Electron. 2, 1066–1075 (1996).
[CrossRef]

J. Biomed. Opt.

D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light forces the pace: optical manipulation for biophotonics,” J. Biomed. Opt. 15, 041503 (2010).
[CrossRef] [PubMed]

J. Chem. Phys.

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, 8825–8831 (1999).
[CrossRef]

J. Global Optim.

R. Storn and R. Price, “Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. 11, 341–359 (1997).
[CrossRef]

J. Mod. Opt.

R. Piestun, B. Spektor, and J. Shamir, “Unconventional light distributions in three-dimensional domains,” J. Mod. Opt. 43, 1495–1507 (1996).
[CrossRef]

J. Opt. A

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. B

T. Čižmár, L. C. Dávila Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B 43, 102001 (2010).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf.

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]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1019–1029 (2003).
[CrossRef]

Nature

V. Garcés-Chávez, D. McGloin, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[CrossRef] [PubMed]

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

New J. Phys.

G. Whyte and J. Courtial, “Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg–Saxton algorithm,” New J. Phys. 7, 117 (2005).
[CrossRef]

Opt. Commun.

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

G. Shabtay, “Three-dimensional beam forming and Ewald’s surfaces,” Opt. Commun. 226, 33–37 (2003).
[CrossRef]

M. T. Valentine, N. R. Guydosh, B. Gutiérrez-Medina, A. N. Fehr, J. O. Andreasson, and S. M. Block, “Precision steering of an optical trap by electro-optic deflection,” Opt. Commun. 33, 599–601 (2008).

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[CrossRef]

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

A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. 133, 7–10 (1997).
[CrossRef]

P. C. Morgensen and J. Glückstad, “Dynamic array generation and pattern formation for optical tweezers,” Opt. Commun. 175, 75–81 (2000).
[CrossRef]

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

Opt. Express

Opt. Lett.

Optik

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

Phys. Chem. Chem. Phys.

J. B. Wills, J. R. Butler, J. Palmer, and J. P. Reid, “Using optical landscapes to control, direct, and isolate aerosol particles,” Phys. Chem. Chem. Phys. 11, 8015–8020 (2009).
[CrossRef] [PubMed]

Phys. Rev. A

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Rev. Sci. Instrum.

C. Mio, T. Gong, A. Terray, and D. W. M. Marr, “Design of a scanning laser optical trap for multiparticle manipulation,” Rev. Sci. Instrum. 71, 2196–2200 (2000).
[CrossRef]

E. R. Dufresne and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optics,” Rev. Sci. Instrum. 69, 1974–1977 (1998).
[CrossRef]

Other

D. L. Andrews, Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Elsevier, 2008).
[PubMed]

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

G. C. Spalding, J. Courtial, and R. Di Leonardo “Holographic Optical Tweezers,” in Structured Light and Its Applications, D. L. Andrews, ed. (Elsevier, 2008), Chap. 6.
[CrossRef]

G. Videen, “Light Scattering from a Sphere Near a Plane Surface,” in Light Scattering from Microstructures, Lecture Notes in Physics Volume 534, F. Moreno and F. González, eds. (Springer, 2000), Chapter 5.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Comparison between computational results (Nmax = 130) and analytical results derived in [34]. All fields are calculated at the z = 0 plane. Analytical gradient force in the (A) x-direction and (B) y-direction. (C) Analytical scattering force in the z-direction. Computationally determined gradient field in the (D) x-direction and (E) y-direction. (F) Computationally determined scattering force in the z-direction. Gradient and scattering forces are negligible in the directions that are not shown. Small errors that appear in the outermost regions of the computational results could be eliminated by increasing Nmax .

Fig. 2
Fig. 2

(A) Target force distribution, F T (r), for the z = 0 plane. (B) Target force distribution repeated at several z planes. The displayed force vectors have been scaled for visualization.

Fig. 3
Fig. 3

(A) The best discovered force distribution shown for the z = 0 plane. (B) A zoom of (A). The displayed force vectors have been scaled for visualization.

Fig. 4
Fig. 4

Particle trajectories where all trajectories have been initiated from the z/λ = −1 plane. The first column is a three-quarter view of the trajectories with three slices of the force field included. The second column is a projection along the positive z-axis with the force field at z/λ = −1 superimposed. All figures show particle trajectories beginning from x/λ = −2 (x = −1.029 μm), x/λ = −1 (x = −0.5145 μm), and x = 0. The first row (AB) corresponds to all initial coordinates with y/λ = −1.5 (y = −0.7717 μm). The second row (C-D) corresponds to initial coordinates with y = 0. The initial point is represented by a circle and the final point by a square. The triangle is a central point and indicates the direction of particle movement.

Fig. 5
Fig. 5

Particle trajectories where all trajectories have been initiated from the z/λ = −1 plane. Each column has the same initial x-coordinates for all trajectories. Trajectories in the first column start at x/λ = −2 (x = −1.029 μm), the second at x/λ = −1 (x = −0.5145 μm), and the third at x = 0. The first row (A-C) corresponds to all initial coordinates with y/λ = −1.5 (y = −0.7717 μm). The second row (D-F) corresponds to initial coordinates with y/λ = −1.0 (y = −0.5145 μm). The third and fourth rows have initial conditions with y/λ = −0.5 (y = −0.2572 μm) and y = 0, respectively. Three slices of the force field are included. The initial point is represented by a red circle and the final point by a red square. The red triangle is a central point and indicates the direction of particle movement.

Fig. 6
Fig. 6

(A)–(F) show the z-component of the Poynting vector at the z/λ = −2, z/λ = −1.4, z/λ = −0.8, z/λ = 0.8, z/λ = 1.4, and z/λ = 2 planes, respectively. (G)–(I) show, respectively, the x-, y-, and z-components of the Poynting vector at the z = 0 plane. (A) and (F) are the only plots of Sz with negative values.

Equations (31)

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E ( r , t ) = R e [ E ( r ) exp ( i ω t ) ]
2 E ( r ) k 2 E ( r ) = 0
E ( r ) = A n = 1 m = n n [ a n m ( 1 ) R g M n m ( k r , θ , φ ) + a n m ( 2 ) R g N n m ( k r , θ , φ ) ]
H ( r ) = A ( ε r ε 0 μ r μ 0 ) 1 / 2 n = 1 m = n n [ a n m ( 1 ) R g N n m ( k r , θ , φ ) + a n m ( 2 ) R g M n m ( k r , θ , φ ) ]
R g M n m ( k r , θ , φ ) = N n j n ( k r ) C n m ( θ , φ )
R g N n m ( k r , θ , φ ) = j n ( k r ) k r N n P n m ( θ , φ ) + N n ( j n 1 ( k r ) n j n ( k r ) k r ) B n m ( θ , φ )
B n m ( θ , φ ) = θ ^ θ Y n m ( θ , φ ) + φ ^ i m sin θ Y n m ( θ , φ )
C n m ( θ , φ ) = θ ^ i m sin θ Y n m ( θ , φ ) φ ^ θ Y n m ( θ , φ )
P n m ( θ , φ ) = r ^ Y n m ( θ , φ )
Y n m ( θ , φ ) = 2 n + 1 4 π ( n m ) ! ( n + m ) ! P n m ( cos θ ) e i m φ
F g r a d ( r , t ) = 4 π n m 2 ε 0 R 3 ( n p 2 n m 2 n p 2 + 2 n m 2 ) 1 2 E 2 ( r , t )
F g r a d ( r ) = F g r a d ( r , t ) = π n m 2 ε 0 R 3 ( n p 2 n m 2 n p 2 + 2 n m 2 ) | E ( r ) | 2
S ( r , t ) E ( r , t ) × H ( r , t )
H ( r , t ) = R e [ H ( r ) exp ( i ω t ) ] .
F s c a t ( r ) = n m c C s c a t S ( r , t )
S ( r , t ) = 1 2 R e [ E * ( r ) × H ( r ) ]
C s c a t = 8 3 π k 4 R 6 ( n p 2 n m 2 n p 2 + 2 n m 2 ) 2 .
F t o t a l ( r ) = F g r a d ( r ) + F s c a t ( r ) .
A 2 = 8 P i k 2 | | a ˜ | | μ m ε m .
p n , 1 ( 1 ) = i n π ( 2 n + 1 ) ( i θ ^ + φ ^ ) e ^ i p n , 1 ( 2 ) = i n π ( 2 n + 1 ) ( i θ ^ + φ ^ ) e ^ i p n , 1 ( 1 ) = i n π ( 2 n + 1 ) ( i θ ^ φ ^ ) e ^ i p n , 1 ( 2 ) = i n π ( 2 n + 1 ) ( i θ ^ + φ ^ ) e ^ i
a ˜ n = g n p ˜ n
s 1 k w 0
g n ( 1 ) = exp [ s 2 ( n 1 ) ( n + 2 ) ] g n ( 3 ) = g n ( 1 ) + exp [ s 2 ( n 1 ) ( n + 2 ) ] ( n 1 ) ( n + 2 ) s 4 [ 3 ( n 1 ) ( n + 2 ) s 2 ] g n ( 5 ) = g n ( 3 ) + exp [ s 2 ( n 1 ) ( n + 2 ) ] × ( n 1 ) 2 ( n + 2 ) 2 s 8 [ 10 5 ( n 1 ) ( n + 2 ) s 2 + 0.5 ( n 1 ) 2 ( n + 2 ) 2 s 4 ] .
F s c a t , z ( a ) ( r ) = z ^ n m c 8 3 π k 4 R 6 ( n p 2 n m 2 n p 2 + 2 n m 2 ) 2 ( 2 P i π w 0 2 ) 1 1 + ( 2 z ˜ ) 2 exp [ 2 ( x ˜ 2 + y ˜ 2 ) 1 + ( 2 z ˜ ) 2 ] .
F g r a d , x ( a ) ( r ) = x ^ 2 π n m R 3 c ( n p 2 n m 2 n p 2 + 2 n m 2 ) 4 x ˜ / w 0 1 + ( 2 z ˜ ) 2 ( 2 P i π w 0 2 ) 1 1 + ( 2 z ˜ ) 2 exp [ 2 ( x ˜ 2 + y ˜ 2 ) 1 + ( 2 z ˜ ) 2 ]
F g r a d , y ( a ) ( r ) = y ^ 2 π n m R 3 c ( n p 2 n m 2 n p 2 + 2 n m 2 ) 4 y ˜ / w 0 1 + ( 2 z ˜ ) 2 ( 2 P i π w 0 2 ) 1 1 + ( 2 z ˜ ) 2 exp [ 2 ( x ˜ 2 + y ˜ 2 ) 1 + ( 2 z ˜ ) 2 ]
F g r a d , z ( a ) ( r ) = z ^ 2 π n m R 3 c ( n p 2 n m 2 n p 2 + 2 n m 2 ) 8 z ˜ / ( k w 0 2 ) 1 + ( 2 z ˜ ) 2 [ 1 2 ( x ˜ 2 + y ˜ 2 ) 1 + ( 2 z ˜ ) 2 ] × ( 2 P i π w 0 2 ) 1 1 + ( 2 z ˜ ) 2 exp [ 2 ( x ˜ 2 + y ˜ 2 1 + ( 2 z ˜ ) 2 ] .
α ( 1 ) = [ a 1 , 1 ( 1 ) , a 1 , 0 ( 1 ) , a 1 , 1 ( 1 ) , a 2 , 2 ( 1 ) , a 2 , 1 ( 1 ) , , a N m a x , N m a x 1 ( 1 ) , a N m a x , N m a x ( 1 ) ]
α ( 2 ) = [ a 1 , 1 ( 2 ) , a 1 , 0 ( 2 ) , a 1 , 1 ( 2 ) , a 2 , 2 ( 2 ) , a 2 , 1 ( 2 ) , , a N m a x , N m a x 1 ( 2 ) , a N m a x , N m a x ( 2 ) ]
v = [ R e { α ( 1 ) } , I m { α ( 1 ) } , R e { α ( 2 ) } , I m { α ( 2 ) } ]
min v r n R ( F t ( r n ; v t ) F T ( r n ) ) 2

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