Abstract

A closed-form analytical expression of the force on an infinite lossless dielectric cylinder due to multiple plane wave incidences is proposed. The formula for a TE polarization is derived and completes our previous work which was limited to TM polarizations. A unified form of the analytical expression of the force is proposed and used to study the curvature of the one dimensional potential of an optical lattice created by the interference of three plane waves. It is shown that the points of zero curvature yield optical vortices which can be used to stably trap particles of particular sizes and index contrasts with the background. Under these circumstances, the trajectories of the particles can be assimilated to spirals whose centers correspond to the points of undetermined phase in the optical landscape.

© 2007 Optical Society of America

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References

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  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  12. J.-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner, and R. Salathé, "Building optical matter with binding and trapping forces," Proc. SPIE 5514, 309-317 (2004).
    [CrossRef]
  13. T. M. Grzegorczyk and J. A. Kong, "Analytical expression of the force due to multiple TM plane wave incidences on an infinite dielectric cylinder," J. Opt. Soc. Am. B 24, 644-652 (2006).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  20. B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, "Optical momentum transfer to absorbing Mie particles," Phys. Rev. Lett. 97, 133902 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]

2006 (6)

2005 (2)

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A. C. Riches, and K. Dholakia, "Light-induced cell separation in a tailored optical landscape," Appl. Phys. Lett. 87, 123901 (2005).
[CrossRef]

D. Cojoc, V. Garbin, E. Ferrari, L. Businaro, F. Romanato, and E. Di Fabrizio, "Laser trapping and micromanipulation using optical vortices," Microelectron. Eng. 78-79, 125-131 (2005).
[CrossRef]

2004 (3)

M. V. Berry, "Optical vortices evolving from helicoidal integer and fractional phase steps," J. Opt. A: Pure Appl. Opt. 6, 259-268 (2004).
[CrossRef]

J.-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner, and R. Salathé, "Building optical matter with binding and trapping forces," Proc. SPIE 5514, 309-317 (2004).
[CrossRef]

P. Zemanek, V. Karasek, and A. Sasso, "Optical forces acting on Rayleigh particle placed into interference field," Opt. Commun. 240, 401-415 (2004).
[CrossRef]

2003 (1)

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

2002 (1)

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

2001 (1)

J. Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001).
[CrossRef]

2000 (1)

P. C. Chaumet and M. Nieto-Vesperinas, "Coupled dipole method determination of the electromagnetic force on particle over a flat dielectric substrate," Phys. Rev. B 61, 14119-14127 (2000).
[CrossRef]

1996 (1)

1992 (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

A. Ashkin, "Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime," Biophys. J. 61, 569-582 (1992).
[CrossRef] [PubMed]

1990 (1)

M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, "Optical Matter: crystallization and binding in intense optical fields," Science 249, 749-754 (1990).
[CrossRef] [PubMed]

1988 (1)

B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333, 848-872 (1988).
[CrossRef]

1987 (1)

J. F. Nye and J. V. Hajnal, "The wave structure of monochromatic electromagnetic radiation," Proc. R. Soc. London Ser. A 409, 21-36 (1987).
[CrossRef]

1986 (1)

Appl. Phys. Lett. (1)

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A. C. Riches, and K. Dholakia, "Light-induced cell separation in a tailored optical landscape," Appl. Phys. Lett. 87, 123901 (2005).
[CrossRef]

Astrophys. J. (1)

B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333, 848-872 (1988).
[CrossRef]

Biophys. J. (1)

A. Ashkin, "Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime," Biophys. J. 61, 569-582 (1992).
[CrossRef] [PubMed]

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

D. Maystre and P. Vincent, "Making photonic crystals using trapping and binding optical forces on particles," J. Opt. A: Pure Appl. Opt. 8, 1059-1066 (2006).
[CrossRef]

M. V. Berry, "Optical vortices evolving from helicoidal integer and fractional phase steps," J. Opt. A: Pure Appl. Opt. 6, 259-268 (2004).
[CrossRef]

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

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

Microelectron. Eng. (1)

D. Cojoc, V. Garbin, E. Ferrari, L. Businaro, F. Romanato, and E. Di Fabrizio, "Laser trapping and micromanipulation using optical vortices," Microelectron. Eng. 78-79, 125-131 (2005).
[CrossRef]

Nature (1)

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

Opt. Commun. (3)

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

J. Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001).
[CrossRef]

P. Zemanek, V. Karasek, and A. Sasso, "Optical forces acting on Rayleigh particle placed into interference field," Opt. Commun. 240, 401-415 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Phys. Rev. B (1)

P. C. Chaumet and M. Nieto-Vesperinas, "Coupled dipole method determination of the electromagnetic force on particle over a flat dielectric substrate," Phys. Rev. B 61, 14119-14127 (2000).
[CrossRef]

Phys. Rev. Lett. (2)

T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, "Stable optical trapping based on optical binding forces," Phys. Rev. Lett. 96, 113903 (2006).
[CrossRef] [PubMed]

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, "Optical momentum transfer to absorbing Mie particles," Phys. Rev. Lett. 97, 133902 (2006).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A (1)

J. F. Nye and J. V. Hajnal, "The wave structure of monochromatic electromagnetic radiation," Proc. R. Soc. London Ser. A 409, 21-36 (1987).
[CrossRef]

Proc. SPIE (1)

J.-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner, and R. Salathé, "Building optical matter with binding and trapping forces," Proc. SPIE 5514, 309-317 (2004).
[CrossRef]

Science (1)

M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, "Optical Matter: crystallization and binding in intense optical fields," Science 249, 749-754 (1990).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1.
Fig. 1.

Configuration of the problem: an infinitely long cylinder of permittivity εp is embedded in a background of permittivity ε and is subject to multiple in-plane incidences.

Fig. 2.
Fig. 2.

Curvature of the pseudo-potential along the axis at ρ 0 = (0,0) and ρ 1 = (2λ/(3√3ε), 0) a (ρ 0 corresponds to the point of maximum field intensity while ρ 1, shown by the right-most ‘+’ sign in Fig. 3, corresponds to the point of minimum field intensity). The incident field is obtained from three plane waves of equal amplitude and directions separated by 2π/3, either TE or TM polarized. The inset depicts a zoomed version of the black rectangle and shows the identical crossing points for the respective polarizations. Other parameters: λ = 1064 nm, ε= 1.69ε 0, εp = 2.56ε 0.

Fig. 3.
Fig. 3.

Background pattern left: phase of the incident electric field (in degrees). Background pattern right: incident TM intensity distribution (in V 2/m 2). Arrows (symmetric in x, only x > 0 is shown for clarity): force on a particle with parameters a = 0.2264λ, λ = 1064 nm, ε = 1.69ε 0, εp = 2.56ε 0. The ‘+’ signs indicate the minimum field intensity points corresponding to possible optical vortices (the right-most sign corresponds to ρ 1). Trajectory ‘1’ corresponds to a particle with a = 0.243λ starting at (-200 nm, 380 nm) (denoted by the ‘x’ sign), and shows that the particle is not trapped by the landscape. Trajectory ‘2’ corresponds to a particle with a = 0.2264λ starting at (-200 nm, -380 nm) (denoted by the ‘x’ sign), and shows that the particle is trapped by the landscape in a spiral-like manner. Dimensions along the horizontal axes are given in μm and the two axes are equal.

Fig. 4.
Fig. 4.

Normalized normal component of the force along the white circle (centered around ρ 1) for two sizes of particles. The always negative values for a = 0.243λ indicate that the normal force is always attracting the particle toward ρ 1, yielding a spiral attractor.

Equations (44)

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F TE = x ̂ aπε 2 ( n ( n ( n + 1 ) ( ka ) 2 1 ) H z n H z n + 1 * E ϕ n E ϕ n + 1 * )
+ y ̂ aπε 2 ( n ( n ( n + 1 ) ( ka ) 2 1 ) H z n H z n + 1 * E ϕ n E ϕ n + 1 * )
+ x ̂ πε 2 k ( n n H z n ( E ϕ n + 1 * E ϕ n 1 * ) )
- y ̂ πε 2 k ( n n H z n ( E ϕ n + 1 * + E ϕ n 1 * ) ) ,
E ϕ n = k [ a n ( M ) s H n ( 1 ) ( k ρ ) + a n ( M ) J n ( k ρ ) ]
H z n = k [ a n ( M ) s H n ( 1 ) ( k ρ ) + a n ( M ) J n ( k ρ ) ]
a n ( M ) = i n + 1 e in ϕ i k E hi ,
E ϕ n = 2 k iπa J n ( k p a ) D n TE , H z n = 2 k p iπa J n ( k p a ) D n TE ,
D n TE = k H n ( 1 ) ( ka ) J n ( k p a ) k p H n ( 1 ) ( ka ) J n ( k p a ) .
A = ( 2 πa ) 2 i k 2 E hi 2 n e i ϕ i D n TE D n + 1 T E * ×
[ ( ( k p 2 k 2 + k 2 k p 2 ) n ( n + 1 ) a k p 2 a + k 2 a ) J n ( k p a ) J n + 1 ( k p a )
k 2 k p ( n + 1 ) J n + 1 2 ( k p a ) k 2 k p n J n 2 ( k p a ) ]
B = ( 2 πa ) 2 i k 2 E hi 2 n [ e i ϕ i k p n J n 2 ( k p a ) D n TE D n + 1 T E *
e i ϕ i k p ( n + 1 ) J n + 1 2 ( k p a ) D n T E * D n + 1 TE + n ( n + 1 ) a J n ( k p a ) J n + 1 ( k p a ) ×
( e i ϕ i D n T E * D n + 1 TE e i ϕ i D n TE D n + 1 T E * ) ]
F TE = k ̂ i 4 ε πa k p 2 k 2 k 2 E hi 2 0 + ( D n T E * D n + 1 TE ) ×
n ( n + 1 ) ( ka ) 2 J n ( k p a ) J n + 1 ( k p a ) + J n ( k p a ) J n + 1 ( k p a ) D n TE 2 D n + 1 TE 2
F ij TE = ( x ̂ cos Φ ij + y ̂ sin Φ ij ) 4 ε πa k p 2 k 2 k 2 E hi 2
0 + ( D n T E * D n + 1 TE e i ( n + 0.5 ) ( ϕ i ϕ j ) ) ×
n ( n + 1 ) ( ka ) 2 J n ( k p a ) J n + 1 ( k p a ) + J n ( k p a ) J n + 1 ( k p a ) D n TE 2 D n + 1 TE 2
F ( p ) = i , j = 1 M F ij ( p ) ,
F ij ( p ) = e i Φ ij K ( p ) n = 0 + Λ ( p ) ( γ n ( p ) β n ) ,
K ( p ) = 4 ε πa k p 2 k 2 k 2 E ( p ) 2 ,
Φ ij = ( k i k j ) ρ + ϕ i + ϕ j 2 ,
β n = e i ( n + 0.5 ) ( ϕ i ϕ j ) ,
γ n ( p ) = D n ( p ) * D n + 1 ( p ) ,
Λ TM = J n ( K p a ) J n + 1 ( k p a ) D n TM 2 D n + 1 TM 2 ,
Λ TE = n ( n + 1 ) ( ka ) 2 J n ( k p a ) J n + 1 ( k p a ) + J n ( k p a ) J n + 1 ( k p a ) D n TE 2 D n + 1 TE 2 ,
D n TE = k H n ( 1 ) ( ka ) J n ( k p a ) k p H n ( 1 ) ( ka ) J n ( k p a ) ,
D n TE = k H n ( 1 ) ( ka ) J n ( k p a ) k p H n ( 1 ) ( ka ) J n ( k p a ) .
F ij ( p ) + F ji ( p ) = 2 K ( p ) e i ( ϕ i + ϕ j ) 2
n = 0 + Λ ( p ) [ ( α ) ( β n ) ( γ n ( p ) ) + i ( α ) ( β n ) ( γ n ( p ) ) ] .
2 U ( p ) x 2 = F ( p ) x ,
2 U ( p ) x 2 = 2 i k i j x K ( p ) e i ( ϕ i + ϕ j ) 2 i = 1 2 j = i + 1 3 n = 0 + Λ ( p ) .
[ cos ( k ij ρ ) ( β n ) ( γ n ( p ) ) + i sin ( k ij ρ ) ( β n ) ( γ n ( p ) ) ]
e i ϕ i e in ϕ i e i ( n + 1 ) ϕ j ,
A = ( 2 πa ) 2 2 a k 2 E hi 2 e ( ϕ i + ϕ j ) 2 n = 0 + 1 D n TE 2 D n + 1 TE 2 ×
[ n ( n + 1 ) ( ka ) 2 k p 2 J n J n + 1 ( k p 2 J n J n + 1 + K 2 J n J n + 1 ) ] ×
( D n T E * D n + 1 TE e i ( n + 0.5 ) ( ϕ i ϕ j ) ) ,
B = ( 2 πa ) 2 2 i k p k 2 E hi 2 cos ( ϕ i + ϕ j 2 ) ×
n e i ( n + 0.5 ) ( ϕ i ϕ j ) D n TE D n + 1 T E * n J n J n + 1 ,
A = ( 2 πa ) 2 i a k 2 E hi 2 e i ϕ i ×
n 1 D n T E * D n + 1 TE [ ( n ( n + 1 ) ( ka ) 2 1 ) k p 2 J n J n + 1 k 2 J n J n + 1 ]
B = ( 2 πa ) 2 2 i k p k 2 E hi 2 cos ϕ i n n J n J n + 1 D n TE D n + 1 T E * .

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