Abstract

Radiation forces exerted upon a dielectric, circular-shaped cylinder of infinite length illuminated by a non-paraxial cylindrical Gaussian beam are considered. Vectorial projections of the radiation pressure force on a dielectric, arbitrary- and circular-shaped cylinder are expressed analytically. In particular, the radiation force is expressed through coefficients of the decomposition of the non-paraxial Gaussian beam into the cylindrical functions. Using numerical examples, a possibility to optically trap a circular-shaped cylinder in a non-paraxial cylindrical Gaussian beam is demonstrated.

© 2006 Optical Society of America

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References

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  1. G.   Gouesbet, B.   Maheu, and G.   Grehan, "Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation," J. Opt. Soc. Am A 5, 1427-1443 (1988).
    [CrossRef]
  2. G.   Gouesbet and J. A. Lock, "A rigorous justification of the localized approximation to the beam-shape coefficients in the generalized Lorenz-Mie theory II. Off-axis beams," J. Opt. Soc. Am A 2, 2516-2525 (1994).
    [CrossRef]
  3. F. Ren, G.   Grehan, and G.   Gouesbet, "Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorenz-Mie theory and associated resonance effects," Opt. Commun. 108, 343-354 (1994).
    [CrossRef]
  4. G. Martinet-Lagarde, B. Pouligny, M. A. Angelova, G. Grehan, and G. Gouesbet, "Trapping and levitation of a dielectric sphere with off-centered Gaussian beams, II-GLMT analysis," Pure Appl. Opt. 4, 571-585 (1995).
    [CrossRef]
  5. K. F. Ren, G. Grehan, and G. Gouesbet, "Prediction of reverse radiation pressure by generalized Lorenz-Mie theory," Appl. Opt. 35, 2702-2710 (1996).
    [CrossRef] [PubMed]
  6. H. Polaert, G. Grehan, and G. Gouesbet, "Improved standard beams with applications to reverse radiation pressure," Appl. Opt. 37, 2435-2440 (1998).
    [CrossRef]
  7. H. Polaert, G. Grehan, and G. Gouesbet, "Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam," Opt. Commun 155, 169-179 (1998).
    [CrossRef]
  8. G. Gouesbet, "Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz-Mie theory for spheres," J. Opt. Soc. Am A 16, 1641-1650 (1999).
    [CrossRef]
  9. J.   Barton, D.   Alexander, and S.   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]
  10. R. Gussgard, T.   Lindmo, and I.   Brevik, "Calculation of the trapping force in a strongly focused laser beam," J. Opt. Soc. Am B 9, 1922-1930 (1992).
    [CrossRef]
  11. A. Rohrbach and E. H. K.  Stelzer, "Optical trapping of a dielectric particle in arbitrary fields," J. Opt. Soc. Am A 18, 839-853 (2001).
    [CrossRef]
  12. A. Rohrbach, and E. H. K. Stelzer, "Trapping forces, force constant, and potential depths for dielectric spheres in the presence of spherical aberration," Appl. Opt. 41, 2494-2507 (2002).
    [CrossRef] [PubMed]
  13. J. A. Lock, "Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. I. Localized model description of an on-axis tightly focused laser beam with spherical aberration," Appl. Opt. 43, 2532-2544 (2004).
    [CrossRef] [PubMed]
  14. J. A. Lock, "Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. II. On-axis trapping force," Appl. Opt. 43, 2545-2554 (2004).
    [CrossRef] [PubMed]
  15. D.  Ganic, X.  Gan, and M. Gu, "Exact radiation trapping force calculation based on vectorial diffraction theory," Opt. Express 12, 2670-2675 (2004).
    [CrossRef] [PubMed]
  16. T. A.   Nieminen, N. R.   Heckenberg, and H.   Rubinstein-Dunlop, "Computational modeling of optical tweezers," in Optical Trapping and Optical Micromanipulation; K. Dholakia, and G. C. Spalding, eds., Proc. SPIE 5514, 514-523 (2004).
    [CrossRef]
  17. A. Mazolli, P. A.  Maia Neto, and H. M.  Nussenzveig. "Theory of trapping forces in optical tweezers," Proc. R. Soc. London 459, 3021-3041 (2003).
    [CrossRef]
  18. Y. K.   Nahmias, and D. J. Oddl. "Analysis of radiation forces in laser trapping and laser-guided direct writing application," IEEE J. Qauntum. Electron., 38-2, 1-10 (2002).
  19. R.   Pobre, and C.   Saloma. "Radiation forces on nonlinear microsphere by a tightly focused Gaussian beam," Appl. Opt., 41-36, 7694-7701 (2002).
    [CrossRef]
  20. P. L.  Marston, and J. H.  Crichton. "Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave," Phys. Rev. A., 30-5, 2508-2516 (1984).
    [CrossRef]
  21. E. Zimmerman, R. Dandliner, and N. Souli. "Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach," J. Opt. Soc. Am. A,  12, 398-403 (1995).
    [CrossRef]
  22. Z. Wu, and L. Guo. "Electromagnetic scattering from a multilayered cylinder arbitrarily located in a Gaussian beam, a new recursive algorithm," Prog. Electromagn. Res. 18, 317-333, (1998).
    [CrossRef]
  23. L. Mees, K. F. Ren, G. Grehan, and G. Gouesbet. "Scattering of a Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation, numerical results," Appl. Opt.,  38, 1867-1876 (1999).
    [CrossRef]
  24. L. D. Landau, and E. M. Lifshitz, "Brief course in theoretical physics. Mechanics. Electrodynamics," (Moscow, Nauka Publishers, Book 1, 1969).

2004

2003

A. Mazolli, P. A.  Maia Neto, and H. M.  Nussenzveig. "Theory of trapping forces in optical tweezers," Proc. R. Soc. London 459, 3021-3041 (2003).
[CrossRef]

2002

2001

A. Rohrbach and E. H. K.  Stelzer, "Optical trapping of a dielectric particle in arbitrary fields," J. Opt. Soc. Am A 18, 839-853 (2001).
[CrossRef]

1999

G. Gouesbet, "Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz-Mie theory for spheres," J. Opt. Soc. Am A 16, 1641-1650 (1999).
[CrossRef]

L. Mees, K. F. Ren, G. Grehan, and G. Gouesbet. "Scattering of a Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation, numerical results," Appl. Opt.,  38, 1867-1876 (1999).
[CrossRef]

1998

H. Polaert, G. Grehan, and G. Gouesbet, "Improved standard beams with applications to reverse radiation pressure," Appl. Opt. 37, 2435-2440 (1998).
[CrossRef]

Z. Wu, and L. Guo. "Electromagnetic scattering from a multilayered cylinder arbitrarily located in a Gaussian beam, a new recursive algorithm," Prog. Electromagn. Res. 18, 317-333, (1998).
[CrossRef]

H. Polaert, G. Grehan, and G. Gouesbet, "Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam," Opt. Commun 155, 169-179 (1998).
[CrossRef]

1996

1995

G. Martinet-Lagarde, B. Pouligny, M. A. Angelova, G. Grehan, and G. Gouesbet, "Trapping and levitation of a dielectric sphere with off-centered Gaussian beams, II-GLMT analysis," Pure Appl. Opt. 4, 571-585 (1995).
[CrossRef]

E. Zimmerman, R. Dandliner, and N. Souli. "Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach," J. Opt. Soc. Am. A,  12, 398-403 (1995).
[CrossRef]

1994

G.   Gouesbet and J. A. Lock, "A rigorous justification of the localized approximation to the beam-shape coefficients in the generalized Lorenz-Mie theory II. Off-axis beams," J. Opt. Soc. Am A 2, 2516-2525 (1994).
[CrossRef]

F. Ren, G.   Grehan, and G.   Gouesbet, "Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorenz-Mie theory and associated resonance effects," Opt. Commun. 108, 343-354 (1994).
[CrossRef]

1992

R. Gussgard, T.   Lindmo, and I.   Brevik, "Calculation of the trapping force in a strongly focused laser beam," J. Opt. Soc. Am B 9, 1922-1930 (1992).
[CrossRef]

1989

J.   Barton, D.   Alexander, and S.   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]

1988

G.   Gouesbet, B.   Maheu, and G.   Grehan, "Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation," J. Opt. Soc. Am A 5, 1427-1443 (1988).
[CrossRef]

Alexander, D.

J.   Barton, D.   Alexander, and S.   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]

Angelova, M. A.

G. Martinet-Lagarde, B. Pouligny, M. A. Angelova, G. Grehan, and G. Gouesbet, "Trapping and levitation of a dielectric sphere with off-centered Gaussian beams, II-GLMT analysis," Pure Appl. Opt. 4, 571-585 (1995).
[CrossRef]

Barton, J.

J.   Barton, D.   Alexander, and S.   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]

Brevik, I.

R. Gussgard, T.   Lindmo, and I.   Brevik, "Calculation of the trapping force in a strongly focused laser beam," J. Opt. Soc. Am B 9, 1922-1930 (1992).
[CrossRef]

Dandliner, R.

Gan, X.

Ganic, D.

Gouesbet, G.

G. Gouesbet, "Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz-Mie theory for spheres," J. Opt. Soc. Am A 16, 1641-1650 (1999).
[CrossRef]

L. Mees, K. F. Ren, G. Grehan, and G. Gouesbet. "Scattering of a Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation, numerical results," Appl. Opt.,  38, 1867-1876 (1999).
[CrossRef]

H. Polaert, G. Grehan, and G. Gouesbet, "Improved standard beams with applications to reverse radiation pressure," Appl. Opt. 37, 2435-2440 (1998).
[CrossRef]

H. Polaert, G. Grehan, and G. Gouesbet, "Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam," Opt. Commun 155, 169-179 (1998).
[CrossRef]

K. F. Ren, G. Grehan, and G. Gouesbet, "Prediction of reverse radiation pressure by generalized Lorenz-Mie theory," Appl. Opt. 35, 2702-2710 (1996).
[CrossRef] [PubMed]

G. Martinet-Lagarde, B. Pouligny, M. A. Angelova, G. Grehan, and G. Gouesbet, "Trapping and levitation of a dielectric sphere with off-centered Gaussian beams, II-GLMT analysis," Pure Appl. Opt. 4, 571-585 (1995).
[CrossRef]

F. Ren, G.   Grehan, and G.   Gouesbet, "Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorenz-Mie theory and associated resonance effects," Opt. Commun. 108, 343-354 (1994).
[CrossRef]

G.   Gouesbet and J. A. Lock, "A rigorous justification of the localized approximation to the beam-shape coefficients in the generalized Lorenz-Mie theory II. Off-axis beams," J. Opt. Soc. Am A 2, 2516-2525 (1994).
[CrossRef]

G.   Gouesbet, B.   Maheu, and G.   Grehan, "Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation," J. Opt. Soc. Am A 5, 1427-1443 (1988).
[CrossRef]

Grehan, G.

L. Mees, K. F. Ren, G. Grehan, and G. Gouesbet. "Scattering of a Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation, numerical results," Appl. Opt.,  38, 1867-1876 (1999).
[CrossRef]

H. Polaert, G. Grehan, and G. Gouesbet, "Improved standard beams with applications to reverse radiation pressure," Appl. Opt. 37, 2435-2440 (1998).
[CrossRef]

H. Polaert, G. Grehan, and G. Gouesbet, "Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam," Opt. Commun 155, 169-179 (1998).
[CrossRef]

K. F. Ren, G. Grehan, and G. Gouesbet, "Prediction of reverse radiation pressure by generalized Lorenz-Mie theory," Appl. Opt. 35, 2702-2710 (1996).
[CrossRef] [PubMed]

G. Martinet-Lagarde, B. Pouligny, M. A. Angelova, G. Grehan, and G. Gouesbet, "Trapping and levitation of a dielectric sphere with off-centered Gaussian beams, II-GLMT analysis," Pure Appl. Opt. 4, 571-585 (1995).
[CrossRef]

F. Ren, G.   Grehan, and G.   Gouesbet, "Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorenz-Mie theory and associated resonance effects," Opt. Commun. 108, 343-354 (1994).
[CrossRef]

G.   Gouesbet, B.   Maheu, and G.   Grehan, "Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation," J. Opt. Soc. Am A 5, 1427-1443 (1988).
[CrossRef]

Gu, M.

Guo, L.

Z. Wu, and L. Guo. "Electromagnetic scattering from a multilayered cylinder arbitrarily located in a Gaussian beam, a new recursive algorithm," Prog. Electromagn. Res. 18, 317-333, (1998).
[CrossRef]

Gussgard, R.

R. Gussgard, T.   Lindmo, and I.   Brevik, "Calculation of the trapping force in a strongly focused laser beam," J. Opt. Soc. Am B 9, 1922-1930 (1992).
[CrossRef]

Lindmo, T.

R. Gussgard, T.   Lindmo, and I.   Brevik, "Calculation of the trapping force in a strongly focused laser beam," J. Opt. Soc. Am B 9, 1922-1930 (1992).
[CrossRef]

Lock, J. A.

Maheu, B.

G.   Gouesbet, B.   Maheu, and G.   Grehan, "Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation," J. Opt. Soc. Am A 5, 1427-1443 (1988).
[CrossRef]

Maia Neto, P. A.

A. Mazolli, P. A.  Maia Neto, and H. M.  Nussenzveig. "Theory of trapping forces in optical tweezers," Proc. R. Soc. London 459, 3021-3041 (2003).
[CrossRef]

Martinet-Lagarde, G.

G. Martinet-Lagarde, B. Pouligny, M. A. Angelova, G. Grehan, and G. Gouesbet, "Trapping and levitation of a dielectric sphere with off-centered Gaussian beams, II-GLMT analysis," Pure Appl. Opt. 4, 571-585 (1995).
[CrossRef]

Mazolli, A.

A. Mazolli, P. A.  Maia Neto, and H. M.  Nussenzveig. "Theory of trapping forces in optical tweezers," Proc. R. Soc. London 459, 3021-3041 (2003).
[CrossRef]

Mees, L.

Nussenzveig, H. M.

A. Mazolli, P. A.  Maia Neto, and H. M.  Nussenzveig. "Theory of trapping forces in optical tweezers," Proc. R. Soc. London 459, 3021-3041 (2003).
[CrossRef]

Polaert, H.

H. Polaert, G. Grehan, and G. Gouesbet, "Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam," Opt. Commun 155, 169-179 (1998).
[CrossRef]

H. Polaert, G. Grehan, and G. Gouesbet, "Improved standard beams with applications to reverse radiation pressure," Appl. Opt. 37, 2435-2440 (1998).
[CrossRef]

Pouligny, B.

G. Martinet-Lagarde, B. Pouligny, M. A. Angelova, G. Grehan, and G. Gouesbet, "Trapping and levitation of a dielectric sphere with off-centered Gaussian beams, II-GLMT analysis," Pure Appl. Opt. 4, 571-585 (1995).
[CrossRef]

Ren, F.

F. Ren, G.   Grehan, and G.   Gouesbet, "Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorenz-Mie theory and associated resonance effects," Opt. Commun. 108, 343-354 (1994).
[CrossRef]

Ren, K. F.

Rohrbach, A.

Schaub, S.

J.   Barton, D.   Alexander, and S.   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]

Souli, N.

Stelzer, E. H. K.

Wu, Z.

Z. Wu, and L. Guo. "Electromagnetic scattering from a multilayered cylinder arbitrarily located in a Gaussian beam, a new recursive algorithm," Prog. Electromagn. Res. 18, 317-333, (1998).
[CrossRef]

Zimmerman, E.

Appl. Opt.

J. Appl. Phys.

J.   Barton, D.   Alexander, and S.   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. Opt. Soc. Am A

G. Gouesbet, "Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz-Mie theory for spheres," J. Opt. Soc. Am A 16, 1641-1650 (1999).
[CrossRef]

A. Rohrbach and E. H. K.  Stelzer, "Optical trapping of a dielectric particle in arbitrary fields," J. Opt. Soc. Am A 18, 839-853 (2001).
[CrossRef]

G.   Gouesbet, B.   Maheu, and G.   Grehan, "Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation," J. Opt. Soc. Am A 5, 1427-1443 (1988).
[CrossRef]

G.   Gouesbet and J. A. Lock, "A rigorous justification of the localized approximation to the beam-shape coefficients in the generalized Lorenz-Mie theory II. Off-axis beams," J. Opt. Soc. Am A 2, 2516-2525 (1994).
[CrossRef]

J. Opt. Soc. Am B

R. Gussgard, T.   Lindmo, and I.   Brevik, "Calculation of the trapping force in a strongly focused laser beam," J. Opt. Soc. Am B 9, 1922-1930 (1992).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun

H. Polaert, G. Grehan, and G. Gouesbet, "Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam," Opt. Commun 155, 169-179 (1998).
[CrossRef]

Opt. Commun.

F. Ren, G.   Grehan, and G.   Gouesbet, "Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorenz-Mie theory and associated resonance effects," Opt. Commun. 108, 343-354 (1994).
[CrossRef]

Opt. Express

Proc. R. Soc. London

A. Mazolli, P. A.  Maia Neto, and H. M.  Nussenzveig. "Theory of trapping forces in optical tweezers," Proc. R. Soc. London 459, 3021-3041 (2003).
[CrossRef]

Prog. Electromagn. Res.

Z. Wu, and L. Guo. "Electromagnetic scattering from a multilayered cylinder arbitrarily located in a Gaussian beam, a new recursive algorithm," Prog. Electromagn. Res. 18, 317-333, (1998).
[CrossRef]

Pure Appl. Opt.

G. Martinet-Lagarde, B. Pouligny, M. A. Angelova, G. Grehan, and G. Gouesbet, "Trapping and levitation of a dielectric sphere with off-centered Gaussian beams, II-GLMT analysis," Pure Appl. Opt. 4, 571-585 (1995).
[CrossRef]

Other

T. A.   Nieminen, N. R.   Heckenberg, and H.   Rubinstein-Dunlop, "Computational modeling of optical tweezers," in Optical Trapping and Optical Micromanipulation; K. Dholakia, and G. C. Spalding, eds., Proc. SPIE 5514, 514-523 (2004).
[CrossRef]

Y. K.   Nahmias, and D. J. Oddl. "Analysis of radiation forces in laser trapping and laser-guided direct writing application," IEEE J. Qauntum. Electron., 38-2, 1-10 (2002).

R.   Pobre, and C.   Saloma. "Radiation forces on nonlinear microsphere by a tightly focused Gaussian beam," Appl. Opt., 41-36, 7694-7701 (2002).
[CrossRef]

P. L.  Marston, and J. H.  Crichton. "Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave," Phys. Rev. A., 30-5, 2508-2516 (1984).
[CrossRef]

L. D. Landau, and E. M. Lifshitz, "Brief course in theoretical physics. Mechanics. Electrodynamics," (Moscow, Nauka Publishers, Book 1, 1969).

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Figures (2)

Fig. 1.
Fig. 1.

The Gaussian beam with focus at (- Z 0,Y 0) falls on a circular cylinder with its center at (0;0).

Fig. 2.
Fig. 2.

The Z-axis (a) and Y-axis (b) projections of the pressure force on a circular cylinder ε=1.2 by the Gaussian beam (medium permittivity is ε 1=1).

Equations (16)

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

t V 1 P i d V + t P 0 i = S 1 σ i k n k d S ,
σ i k = ε 0 ε 1 E 2 + μ μ 0 H 2 2 δ i k ε 0 ε 1 E i E k μ μ 0 H i H k ;
F y = 1 2 S 1 { 1 2 [ μ μ 0 H y 2 ε 0 ε 1 E x 2 μ μ 0 H z 2 ] d z + μ μ 0 Re ( H y H z * ) d y } ,
F z = 1 2 S 1 { 1 2 [ μ μ 0 H z 2 ε 0 ε 1 E x 2 μ μ 0 H y 2 ] d y + μ μ 0 Re ( H z H y * ) d y } ,
E x i ( ρ , φ ) = E 0 ω 0 π λ exp [ k 2 ω 0 2 q 2 4 + i k ( Z 0 p Y 0 q ) + i k r cos ( φ γ ) ] d q ,
E x ( ρ , φ ) = E 0 n = i n C n J n ( k r ) e in φ ,
C n = ω 0 π λ exp [ k 2 ω 0 2 q 2 4 + i k 1 q 2 Z 0 i k q Y 0 in arcsin q ] d q ,
H φ i ( r , φ ) = i H 0 n = i n C n J n ( k r ) e in φ , J n ( k r ) = d d ( k r ) J n ( k r ) ,
H r i ( r , φ ) = H 0 n = i n n C n J n ( k r ) k r e i n φ , H 0 = ε 1 ε 0 μ 0 E 0 .
E x S = E 0 n = i n C n S H n ( 1 ) ( k r ) e i n φ , H φ S = i H 0 n = i n C n S H n ( 1 ) ( k r ) e i n φ ,
H r S = H 0 n = n i n C n S H n ( 1 ) ( k r ) k r e i n φ ,
a n = k 1 J n ( k 1 R ) J n ( k R ) k J n ( k 1 R ) J n ( k R ) k 1 J n ( k 1 R ) H n ( 1 ) ( k R ) k J n ( k 1 R ) H n ( 1 ) ( k R ) ,
F y = i ε 0 ε 1 E 0 2 k n = C n ( C n + 1 * a n + 1 * + C n + 1 * a n +
+ 2 C n + 1 * a n a n + 1 * C n 1 * a n 1 * C n 1 * a n 2 C n 1 * a n a n 1 * ) ,
F z = ε 0 ε 1 E 0 2 k n = C n ( C n + 1 * a n + 1 * + C n + 1 * a n +
+ 2 C n + 1 * a n a n + 1 * + C n 1 * a n 1 * + C n 1 * a n + 2 C n 1 * a n a n 1 * ) .

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