Abstract

Recent advances in dynamic holography have resulted in spatial light modulators capable of producing an almost limitless variety of field distributions from a single incident beam. Holographic assembly is a technique that exploits this capability to generate and control multiple foci that can be used to trap and manipulate nanoparticles. Although the forces associated with conventional optical tweezers are well understood, the effects arising from the more complicated interactions associated with holographic assembly are not. We present a general and flexible method, based on T matrix theory, for investigating these effects and use it to calculate the forces between particles in a variety of optical environments.

© 2006 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]

2005 (3)

2004 (1)

K. C. Neuman and S. M. Block, "Optical trapping," Rev. Sci. Instrum. 75, 2787-2809 (2004).
[CrossRef]

2003 (2)

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

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Multipole expansion of strongly focused laser beams," J. Quant. Spectrosc. Radiat. Transf. 79, 1005-1017 (2003).
[CrossRef]

2002 (1)

J. E. Molloy and M. J. Padgett, "Lights, action: optical tweezers," Contemp. Phys. 43, 241-258 (2002).
[CrossRef]

2001 (2)

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, "Numerical modelling of optical trapping," Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Calculation and optical measurement of laser trapping forces on non-spherical particles," J. Quant. Spectrosc. Radiat. Transf. 70, 627-637 (2001).
[CrossRef]

2000 (4)

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]

P. C. Mogensen and J. Glückstad, "Dynamic array generation and pattern formation for optical tweezers," Opt. Commun. 175, 75-81 (2000).
[CrossRef]

A. D. White, "Vector finite element modelling of optical tweezers," Comput. Phys. Commun. 128, 558-564 (2000).
[CrossRef]

J. M. Fernandez-Varea and R. Garcia-Molina, "Hamaker constants of systems involving water obtained from a dielectric function that fulfills the F sum rule," J. Colloid Interface Sci. 231, 394-397 (2000).
[CrossRef] [PubMed]

1999 (1)

K. Okamoto and S. Kawata, "Radiation force exerted on subwavelength particles near a nanoaperture," Phys. Rev. Lett. 83, 4534-4537 (1999).
[CrossRef]

1997 (1)

L. Novotny, R. X. Bian, and X. S. Xie, "Theory of nanometric optical tweezers," Phys. Rev. Lett. 79, 645-648 (1997).
[CrossRef]

1996 (1)

1995 (2)

1994 (2)

1989 (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

1987 (1)

B. Maheu, G. Grehan, and G. Gouesbet, "Laser beam scattering by individual spherical particles: numerical results and application to particle sizing," Part. Charact. 4, 141-146 (1987).
[CrossRef]

1986 (1)

1970 (1)

A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

Ashkin, A.

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

Bian, R. X.

L. Novotny, R. X. Bian, and X. S. Xie, "Theory of nanometric optical tweezers," Phys. Rev. Lett. 79, 645-648 (1997).
[CrossRef]

Bishop, A. I.

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, "Numerical modelling of optical trapping," Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

Bjorkholm, J. E.

Block, S. M.

K. C. Neuman and S. M. Block, "Optical trapping," Rev. Sci. Instrum. 75, 2787-2809 (2004).
[CrossRef]

Brenner, H.

J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics: with Special Applications to Particulate Media, 2nd ed. (Noordhoff, 1973).

Chu, S.

Deng, J. L.

Dziedzic, J. M.

Fernandez-Varea, J. M.

J. M. Fernandez-Varea and R. Garcia-Molina, "Hamaker constants of systems involving water obtained from a dielectric function that fulfills the F sum rule," J. Colloid Interface Sci. 231, 394-397 (2000).
[CrossRef] [PubMed]

Fuller, K. A.

Garcia-Molina, R.

J. M. Fernandez-Varea and R. Garcia-Molina, "Hamaker constants of systems involving water obtained from a dielectric function that fulfills the F sum rule," J. Colloid Interface Sci. 231, 394-397 (2000).
[CrossRef] [PubMed]

Gauthier, R. C.

Glückstad, J.

P. C. Mogensen and J. Glückstad, "Dynamic array generation and pattern formation for optical tweezers," Opt. Commun. 175, 75-81 (2000).
[CrossRef]

Gouesbet, G.

Grehan, G.

G. Gouesbet, J. A. Lock, and G. Grehan, "Partial wave representations of laser beams for use in light scattering calculations," Appl. Opt. 34, 2133-2143 (1995).
[CrossRef] [PubMed]

B. Maheu, G. Grehan, and G. Gouesbet, "Laser beam scattering by individual spherical particles: numerical results and application to particle sizing," Part. Charact. 4, 141-146 (1987).
[CrossRef]

Grier, D. G.

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

Hafner, C.

C. Hafner, The Generalized Multiple Multipole Technique for Computational Electrodynamics (Artech House, 1990).

Haist, T.

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]

Happel, J.

J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics: with Special Applications to Particulate Media, 2nd ed. (Noordhoff, 1973).

Heckenberg, N. R.

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Multipole expansion of strongly focused laser beams," J. Quant. Spectrosc. Radiat. Transf. 79, 1005-1017 (2003).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, "Numerical modelling of optical trapping," Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Calculation and optical measurement of laser trapping forces on non-spherical particles," J. Quant. Spectrosc. Radiat. Transf. 70, 627-637 (2001).
[CrossRef]

Israelachvili, J. N.

J. N. Israelachvili, Intermolecular and Surface Forces, 2nd ed. (Academic, 1992).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975).

Kawata, S.

K. Okamoto and S. Kawata, "Radiation force exerted on subwavelength particles near a nanoaperture," Phys. Rev. Lett. 83, 4534-4537 (1999).
[CrossRef]

Kong, J. A.

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

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

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]

Lock, J. A.

Mackowski, D. W.

Mahanty, J.

J. Mahanty and B. W. Ninham, Dispersion Forces (Academic, 1976).

Maheu, B.

B. Maheu, G. Grehan, and G. Gouesbet, "Laser beam scattering by individual spherical particles: numerical results and application to particle sizing," Part. Charact. 4, 141-146 (1987).
[CrossRef]

Mazo, R. M.

R. M. Mazo, Brownian Motion: Fluctuations, Dynamics and Applications (Clarendon, 2002).

Melia, F.

F. Melia, Electrodynamics (University of Chicago Press, 2001).

Mishchenko, M. I.

D. W. Mackowski and M. I. Mishchenko, "Calculation of the T matrix and the scattering matrix for ensembles of spheres," J. Opt. Soc. Am. A 13, 2266-2278 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Mogensen, P. C.

P. C. Mogensen and J. Glückstad, "Dynamic array generation and pattern formation for optical tweezers," Opt. Commun. 175, 75-81 (2000).
[CrossRef]

Moine, O.

Molloy, J. E.

J. E. Molloy and M. J. Padgett, "Lights, action: optical tweezers," Contemp. Phys. 43, 241-258 (2002).
[CrossRef]

Neuman, K. C.

K. C. Neuman and S. M. Block, "Optical trapping," Rev. Sci. Instrum. 75, 2787-2809 (2004).
[CrossRef]

Nieminen, T. A.

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Multipole expansion of strongly focused laser beams," J. Quant. Spectrosc. Radiat. Transf. 79, 1005-1017 (2003).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, "Numerical modelling of optical trapping," Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Calculation and optical measurement of laser trapping forces on non-spherical particles," J. Quant. Spectrosc. Radiat. Transf. 70, 627-637 (2001).
[CrossRef]

Ninham, B. W.

J. Mahanty and B. W. Ninham, Dispersion Forces (Academic, 1976).

Novotny, L.

L. Novotny, R. X. Bian, and X. S. Xie, "Theory of nanometric optical tweezers," Phys. Rev. Lett. 79, 645-648 (1997).
[CrossRef]

Okamoto, K.

K. Okamoto and S. Kawata, "Radiation force exerted on subwavelength particles near a nanoaperture," Phys. Rev. Lett. 83, 4534-4537 (1999).
[CrossRef]

Padgett, M. J.

J. E. Molloy and M. J. Padgett, "Lights, action: optical tweezers," Contemp. Phys. 43, 241-258 (2002).
[CrossRef]

Reicherter, M.

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]

Rubinsztein-Dunlop, H.

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Multipole expansion of strongly focused laser beams," J. Quant. Spectrosc. Radiat. Transf. 79, 1005-1017 (2003).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, "Numerical modelling of optical trapping," Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Calculation and optical measurement of laser trapping forces on non-spherical particles," J. Quant. Spectrosc. Radiat. Transf. 70, 627-637 (2001).
[CrossRef]

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

Shin, R. T.

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

Stout, B.

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]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Tsang, L.

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

van der Hulst, H. C.

H. C. van der Hulst, Light Scattering from Small Particles (Wiley, 1957).

Wang, Y. Z.

Wei, Q.

White, A. D.

A. D. White, "Vector finite element modelling of optical tweezers," Comput. Phys. Commun. 128, 558-564 (2000).
[CrossRef]

Xie, X. S.

L. Novotny, R. X. Bian, and X. S. Xie, "Theory of nanometric optical tweezers," Phys. Rev. Lett. 79, 645-648 (1997).
[CrossRef]

Xu, Y.-L.

Appl. Opt. (2)

Comput. Phys. Commun. (2)

A. D. White, "Vector finite element modelling of optical tweezers," Comput. Phys. Commun. 128, 558-564 (2000).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, "Numerical modelling of optical trapping," Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

Contemp. Phys. (1)

J. E. Molloy and M. J. Padgett, "Lights, action: optical tweezers," Contemp. Phys. 43, 241-258 (2002).
[CrossRef]

J. Appl. Phys. (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

J. Colloid Interface Sci. (1)

J. M. Fernandez-Varea and R. Garcia-Molina, "Hamaker constants of systems involving water obtained from a dielectric function that fulfills the F sum rule," J. Colloid Interface Sci. 231, 394-397 (2000).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (3)

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

J. Quant. Spectrosc. Radiat. Transf. (2)

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Calculation and optical measurement of laser trapping forces on non-spherical particles," J. Quant. Spectrosc. Radiat. Transf. 70, 627-637 (2001).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Multipole expansion of strongly focused laser beams," J. Quant. Spectrosc. Radiat. Transf. 79, 1005-1017 (2003).
[CrossRef]

Nature (London) (1)

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

P. C. Mogensen and J. Glückstad, "Dynamic array generation and pattern formation for optical tweezers," Opt. Commun. 175, 75-81 (2000).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Part. Charact. (1)

B. Maheu, G. Grehan, and G. Gouesbet, "Laser beam scattering by individual spherical particles: numerical results and application to particle sizing," Part. Charact. 4, 141-146 (1987).
[CrossRef]

Phys. Rev. Lett. (3)

A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970).
[CrossRef]

K. Okamoto and S. Kawata, "Radiation force exerted on subwavelength particles near a nanoaperture," Phys. Rev. Lett. 83, 4534-4537 (1999).
[CrossRef]

L. Novotny, R. X. Bian, and X. S. Xie, "Theory of nanometric optical tweezers," Phys. Rev. Lett. 79, 645-648 (1997).
[CrossRef]

Rev. Sci. Instrum. (1)

K. C. Neuman and S. M. Block, "Optical trapping," Rev. Sci. Instrum. 75, 2787-2809 (2004).
[CrossRef]

Other (10)

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

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

H. C. van der Hulst, Light Scattering from Small Particles (Wiley, 1957).

C. Hafner, The Generalized Multiple Multipole Technique for Computational Electrodynamics (Artech House, 1990).

J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics: with Special Applications to Particulate Media, 2nd ed. (Noordhoff, 1973).

J. Mahanty and B. W. Ninham, Dispersion Forces (Academic, 1976).

J. N. Israelachvili, Intermolecular and Surface Forces, 2nd ed. (Academic, 1992).

R. M. Mazo, Brownian Motion: Fluctuations, Dynamics and Applications (Clarendon, 2002).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975).

F. Melia, Electrodynamics (University of Chicago Press, 2001).

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Figures (9)

Fig. 1
Fig. 1

Intensity profile in the y z plane for a Gaussian beam propagating in the negative z direction and displaced from the origin by 3 wavelengths for different values of n max (see text). The axes are scaled by the wavenumber k of the beam. n max = ( a ) 15, (b) 20, (c) 25, (d) 30.

Fig. 2
Fig. 2

Convergence characteristics for force calculations involving pairs of spheres, each held within a separate optical trap (see text). The sphere radii ρ and separation σ are given by ρ = 0.1 μ m , σ = 0.3 μ m (solid circles) ρ = 0.1 μ m , σ = 1.2 μ m (diamonds); ρ = 0.25 μ m , σ = 1.2 μ m (open circles).

Fig. 3
Fig. 3

Calculations of force and torque on a silica sphere ( ρ = 0.1 μ m ) in a Gaussian beam propagating in the negative z direction with a waist radius of 0.5 μ m in water. (a) A vector field showing the forces on the particle as a function of position in the beam. The contours show the modulus of the force. The inset ellipse shows the sphere drawn to scale. (b) The x component of torque experienced by the sphere as a function of lateral displacement across the horizontal midline of (a) for a beam power of 0.25 W . (c) The y component of trapping efficiency versus horizontal position across the midline of the beam. (d) The z component of trapping efficiency as a function of vertical position along the beam axis.

Fig. 4
Fig. 4

Vector field plots for forces exerted on silica spheres in equilibrium positions in Gaussian beam optical traps as a consequence of a second sphere of the same radius encroaching on the beam: (a) ρ = 0.1 μ m , x y plane; (b) ρ = 0.25 μ m , x y plane; (c) ρ = 0.1 μ m , y z plane; (d) ρ = 0.25 μ m , y z plane.

Fig. 5
Fig. 5

Line sections taken through the test particle in Fig. 4a parallel to (a) k y and (b) k x showing trapping efficiency as a function of separation between the spheres. (c) A similar line section taken through Fig. 4d parallel to k z . (d) The variation in intensity gradient with axial distance from a 0.25 μ m silica sphere located at the focal point of a Gaussian beam.

Fig. 6
Fig. 6

(a) The trapping efficiency of a silica particle ( ρ = 0.1 μ m ) in the equilibrium position of a Gaussian beam when a second trapped particle approaches, the second beam being similarly polarized and in phase with the first. The refractive index of the second particle varies as indicated. The plot shows the force Q y on particle 1 as a function of separation in the y direction. (b) The additional force Δ Q y due to the presence of a particle in the second trap obtained by subtracting the effect of an empty trap from the curves in (a). (c) The trapping efficiency of the same system versus the refractive index of particle 2 determined at k y = 3.13 . (d) A plot analogous to (a) but for spheres of radius = 0.25 μ m . (e) Vector field showing force on first particle as a function of the position of the second ( ρ = 0.25 μ m , n = 1.45 ).

Fig. 7
Fig. 7

Intensity profiles of two Gaussian beams in close proximity in the plane perpendicular to the beam axis and through the focus. (a)–(c) The beams are in phase, with separation of (a) 2 wavelengths, (b) 1 wavelength, and (c) 0.5 wavelength. (d)–(f) The beams are in antiphase, with separations of (d) 2 wavelengths, (e) 1 wavelength, (f) 0.5 wavelength.

Fig. 8
Fig. 8

Forces on silica particles in the equilibrium positions of twin Gaussian beams for varying relative beam phases δ: (a) ρ = 0.1 μ m , (b) ρ = 0.25 μ m . (c) Vector field showing force on first particle as function of position of second for spheres with radius 0.25 μ m and a relative phase of π radians between the two beams.

Fig. 9
Fig. 9

Forces on pairs of particles in the equilibrium positions of twin Gaussian beams for varying relative beam polarizations Δ: (a) ρ = 0.1 μ m , (b) ρ = 0.25 μ m .

Equations (31)

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F = S T ͇ M ( r ) d S ,
T ͇ M ( r ) = 1 2 R [ ϵ E ( r ) E * ( r ) + μ H ( r ) H * ( r ) 1 2 ( ϵ E ( r ) 2 + μ H ( r ) 2 ) I ͇ ] ,
Γ = S r d S [ T ͇ M ( r ) × r ̂ ] .
E inc ( r ) = n = 1 m = n m = n [ a m n Rg M m n ( k r ) + b m n Rg N m n ( k r ) ] ,
E sca ( r ) = n = 1 m = n m = n [ p m n M m n ( k r ) + q m n N m n ( k r ) ] ,
a ̃ = ( a 1 , 1 , a 0 , 1 , a 1 , 1 , a 2 , 2 , , a 2 , 2 , , a n max , n max , , a n max , n max , b 1 , 1 , , b n max , n max ) .
E inc ( r ) = a ̃ Rg V ( r ) ,
E sca ( r ) = p ̃ V ( r ) .
p ̃ = T a ̃ .
a ̃ tot = i = 1 N R ( α i , β i ) A ( k r i , θ i , ϕ i ) a ̃ ,
E 1 , inc ( r ) = E 2 : 1 , inc ( r ) + E 0 : 1 , inc ( r ) ,
E 2 : 1 , inc ( r ) = [ A ( k r 21 , θ 21 , ϕ 21 ) p ̃ 2 ] Rg V ( k r 21 , θ 21 , ϕ 21 ) .
p ̃ 1 = T 1 [ a ̃ 1 + A ( k r 21 , θ 21 , ϕ 21 ) p ̃ 2 ] ,
p ̃ 1 = [ I T 1 A 21 T 2 A 12 ] 1 [ T 1 + T 1 A 21 T 2 Rg A 12 ] a ̃ 1 .
T pair 1 = [ I T 1 A 21 T 2 A 12 ] 1 [ T 1 + T 1 A 21 T 2 Rg A 12 ] .
E 1 , tot = ( a ̃ 1 + A 21 p ̃ 2 ) Rg V ( r ) + p ̃ 1 V ( r ) , r < r 12 ,
H inc ( r ) = j [ ϵ μ ] 1 2 n = 1 m = n m = n [ b m n Rg M m n ( k r ) + a m n Rg N m n ( k r ) ] ,
H sca ( r ) = j [ ϵ μ ] 1 2 n = 1 m = n m = n [ q m n M m n ( k r ) + p m n N m n ( k r ) ] ,
Q = F c n P ,
F i F i 1 F i 1 0.001 ,
F = 6 π η ν a ,
F = A R 12 D 2 ,
M m n ( k r , θ , ϕ ) = γ m n h n ( 1 ) ( k r ) C m n ( θ , ϕ ) ,
N m n ( k r , θ , ϕ ) = γ m n { n ( n + 1 ) k r h n ( 1 ) ( k r ) P m n ( θ , ϕ ) + 1 k r d d ( k r ) [ k r h n ( 1 ) ( k r ) B m n ( θ , ϕ ) ] } ,
C m n ( θ , ϕ ) = [ θ ̂ j m sin θ P n m ( cos θ ) ϕ ̂ d d θ P n m ( cos θ ) ] exp ( j m ϕ ) ,
P m n ( θ , ϕ ) = r ̂ P n m ( cos θ ) exp ( j m ϕ ) ,
B m n ( θ , ϕ ) = [ θ ̂ d d θ P n m ( cos θ ) + ϕ ̂ j m sin θ P n m ( cos θ ) ] exp ( j m ϕ ) ,
γ m n = [ ( 2 n + 1 ) ( n m ) ! 4 π n ( n + 1 ) ( n + m ) ! ] 1 2 .
a 1 n = ( j ) n + 1 2 k [ 4 π ( 2 n + 1 ) ] 1 2 exp [ s 2 ( n + 1 2 ) 2 ] ,
a 1 n = b 1 n = b 1 n = a 1 n ,
s = 1 k w 0 ,

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