Abstract

It is known that core–shell subwavelength nanoparticles consisting of a dielectric shell and a metallic core can simultaneously support electric and magnetic dipolar resonances, which enhance forward scattering and suppress backward scattering. This creates favorable conditions for optical tractor beam applications. Using the generalized Lorenz–Mie theory and Maxwell stress tensor formulation, we demonstrate how optical pulling forces can be induced and optimized by first-order Bessel beams with appropriate polarization. The transverse stability of the core–shell nanoparticle under ambient damping is also verified by linear stability analysis and dynamical simulation.

© 2014 Optical Society of America

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References

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  1. P. L. Marston, J. Acoust. Soc. Am. 120, 3518 (2006).
    [CrossRef]
  2. P. L. Marston, J. Acoust. Soc. Am. 122, 3162 (2007).
    [CrossRef]
  3. A. Mizrahi, Y. Fainman, Opt. Lett. 35, 3405 (2010).
    [CrossRef]
  4. S. H. Lee, Y. Roichman, D. G. Grier, Opt. Express 18, 6988 (2010).
    [CrossRef]
  5. S. Sukhov, A. Dogariu, Opt. Lett. 35, 3847 (2010).
    [CrossRef]
  6. J. Chen, J. Ng, Z. F. Lin, C. T. Chan, Nat. Photonics 5, 531 (2011).
    [CrossRef]
  7. A. Novitsky, C. W. Qiu, H. Wang, Phys. Rev. Lett. 107, 203601 (2011).
    [CrossRef]
  8. J. Sáenz, Nat. Photonics 5, 514 (2011).
    [CrossRef]
  9. S. Aukhov, A. Dogariu, Phys. Rev. Lett. 107, 203602 (2011).
    [CrossRef]
  10. A. Novitsky, C. W. Qiu, A. Lavrinenko, Phys. Rev. Lett. 109, 023902 (2012).
    [CrossRef]
  11. D. B. Ruffner, D. G. Grier, Phys. Rev. Lett. 109, 163903 (2012).
    [CrossRef]
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    [CrossRef]
  13. O. Brzobohatý, V. Karásek, M. Šiler, L. Chvátal, T. Čižmár, P. Zemánek, Nat. Photonics 7, 123 (2013).
    [CrossRef]
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  19. K. Dholakia, T. Čižmár, Nat. Photonics 5, 335 (2011).
    [CrossRef]
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  23. M. S. Wheeler, J. S. Aitchison, M. Mojahedi, Phys. Rev. B 73, 045105 (2006).
    [CrossRef]
  24. R. P. Dominguez, F. L. Tejeira, R. Marques, J. A. S. Gil, New J. Phys. 13, 123017 (2011).
    [CrossRef]
  25. W. Liu, A. E. Miroshnichenko, D. N. Neshev, Y. S. Kivshar, ACS Nano 6, 5489 (2012).
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2013 (4)

A. Dogariu, S. Sukhov, J. J. Saenz, Nat. Photonics 7, 24 (2013).
[CrossRef]

O. Brzobohatý, V. Karásek, M. Šiler, L. Chvátal, T. Čižmár, P. Zemánek, Nat. Photonics 7, 123 (2013).
[CrossRef]

N. Wang, J. Chen, S. Y. Liu, Z. F. Lin, Phys. Rev. A 87, 063812 (2013).
[CrossRef]

G. Gouesbet, J. A. Lock, Appl. Opt. 52, 897 (2013).
[CrossRef]

2012 (3)

W. Liu, A. E. Miroshnichenko, D. N. Neshev, Y. S. Kivshar, ACS Nano 6, 5489 (2012).
[CrossRef]

A. Novitsky, C. W. Qiu, A. Lavrinenko, Phys. Rev. Lett. 109, 023902 (2012).
[CrossRef]

D. B. Ruffner, D. G. Grier, Phys. Rev. Lett. 109, 163903 (2012).
[CrossRef]

2011 (9)

J. Chen, J. Ng, Z. F. Lin, C. T. Chan, Nat. Photonics 5, 531 (2011).
[CrossRef]

A. Novitsky, C. W. Qiu, H. Wang, Phys. Rev. Lett. 107, 203601 (2011).
[CrossRef]

J. Sáenz, Nat. Photonics 5, 514 (2011).
[CrossRef]

S. Aukhov, A. Dogariu, Phys. Rev. Lett. 107, 203602 (2011).
[CrossRef]

R. P. Dominguez, F. L. Tejeira, R. Marques, J. A. S. Gil, New J. Phys. 13, 123017 (2011).
[CrossRef]

F. M. Fazal, S. M. Block, Nat. Photonics 5, 318 (2011).
[CrossRef]

K. Dholakia, T. Čižmár, Nat. Photonics 5, 335 (2011).
[CrossRef]

M. Padgett, R. Bowman, Nat. Photonics 5, 343 (2011).
[CrossRef]

M. L. Juan, M. Righini, R. Quidant, Nat. Photonics 5, 349 (2011).
[CrossRef]

2010 (5)

2009 (1)

G. Gouesbet, J. Quant. Spectrosc. Radiat. Transfer 110, 1223 (2009).
[CrossRef]

2007 (1)

P. L. Marston, J. Acoust. Soc. Am. 122, 3162 (2007).
[CrossRef]

2006 (2)

P. L. Marston, J. Acoust. Soc. Am. 120, 3518 (2006).
[CrossRef]

M. S. Wheeler, J. S. Aitchison, M. Mojahedi, Phys. Rev. B 73, 045105 (2006).
[CrossRef]

2005 (2)

M. S. Wheeler, J. S. Aitchison, M. Mojahedi, Phys. Rev. B 72, 193103 (2005).
[CrossRef]

J. Ng, Z. F. Lin, C. T. Chan, P. Sheng, Phys. Rev. B 72, 085130 (2005).
[CrossRef]

2003 (1)

D. G. Grier, Nature 424, 810 (2003).
[CrossRef]

1995 (1)

Z. Bouchal, M. Olivik, J. Mod. Opt. 42, 1555 (1995).
[CrossRef]

1994 (1)

T. T. Perkins, D. E. Smith, S. Chu, Science 264, 819 (1994).
[CrossRef]

1986 (1)

1972 (1)

P. B. Johnson, R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

1970 (1)

A. Ashkin, Phys. Rev. Lett. 24, 156 (1970).
[CrossRef]

Aitchison, J. S.

M. S. Wheeler, J. S. Aitchison, M. Mojahedi, Phys. Rev. B 73, 045105 (2006).
[CrossRef]

M. S. Wheeler, J. S. Aitchison, M. Mojahedi, Phys. Rev. B 72, 193103 (2005).
[CrossRef]

Ashkin, A.

Aukhov, S.

S. Aukhov, A. Dogariu, Phys. Rev. Lett. 107, 203602 (2011).
[CrossRef]

Bjorkholm, J. E.

Block, S. M.

F. M. Fazal, S. M. Block, Nat. Photonics 5, 318 (2011).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley and Sons, 1983).

Bouchal, Z.

Z. Bouchal, M. Olivik, J. Mod. Opt. 42, 1555 (1995).
[CrossRef]

Bowman, R.

M. Padgett, R. Bowman, Nat. Photonics 5, 343 (2011).
[CrossRef]

Brzobohatý, O.

O. Brzobohatý, V. Karásek, M. Šiler, L. Chvátal, T. Čižmár, P. Zemánek, Nat. Photonics 7, 123 (2013).
[CrossRef]

Chan, C. T.

J. Chen, J. Ng, Z. F. Lin, C. T. Chan, Nat. Photonics 5, 531 (2011).
[CrossRef]

J. Ng, Z. F. Lin, C. T. Chan, Phys. Rev. Lett. 104, 103601 (2010).
[CrossRef]

J. Ng, Z. F. Lin, C. T. Chan, P. Sheng, Phys. Rev. B 72, 085130 (2005).
[CrossRef]

Chen, J.

N. Wang, J. Chen, S. Y. Liu, Z. F. Lin, Phys. Rev. A 87, 063812 (2013).
[CrossRef]

J. Chen, J. Ng, Z. F. Lin, C. T. Chan, Nat. Photonics 5, 531 (2011).
[CrossRef]

J. Chen, J. Ng, P. Wang, Z. F. Lin, Opt. Lett. 35, 1674 (2010).

Christy, R. W.

P. B. Johnson, R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Chu, S.

Chvátal, L.

O. Brzobohatý, V. Karásek, M. Šiler, L. Chvátal, T. Čižmár, P. Zemánek, Nat. Photonics 7, 123 (2013).
[CrossRef]

Cižmár, T.

O. Brzobohatý, V. Karásek, M. Šiler, L. Chvátal, T. Čižmár, P. Zemánek, Nat. Photonics 7, 123 (2013).
[CrossRef]

K. Dholakia, T. Čižmár, Nat. Photonics 5, 335 (2011).
[CrossRef]

Dholakia, K.

K. Dholakia, T. Čižmár, Nat. Photonics 5, 335 (2011).
[CrossRef]

Dogariu, A.

A. Dogariu, S. Sukhov, J. J. Saenz, Nat. Photonics 7, 24 (2013).
[CrossRef]

S. Aukhov, A. Dogariu, Phys. Rev. Lett. 107, 203602 (2011).
[CrossRef]

S. Sukhov, A. Dogariu, Opt. Lett. 35, 3847 (2010).
[CrossRef]

Dominguez, R. P.

R. P. Dominguez, F. L. Tejeira, R. Marques, J. A. S. Gil, New J. Phys. 13, 123017 (2011).
[CrossRef]

Dziedzic, J. M.

Fainman, Y.

Fazal, F. M.

F. M. Fazal, S. M. Block, Nat. Photonics 5, 318 (2011).
[CrossRef]

Friberg, A. T.

J. Turunen, A. T. Friberg, in Progress in Optics (Elsevier, 2009), Vol. 54, Chap. 1.

Gil, J. A. S.

R. P. Dominguez, F. L. Tejeira, R. Marques, J. A. S. Gil, New J. Phys. 13, 123017 (2011).
[CrossRef]

Gouesbet, G.

G. Gouesbet, J. A. Lock, Appl. Opt. 52, 897 (2013).
[CrossRef]

G. Gouesbet, J. Quant. Spectrosc. Radiat. Transfer 110, 1223 (2009).
[CrossRef]

G. Gouesbet, G. Grehan, Generalized Lorenz–Mie Theories (Springer, 2011).

Grehan, G.

G. Gouesbet, G. Grehan, Generalized Lorenz–Mie Theories (Springer, 2011).

Grier, D. G.

D. B. Ruffner, D. G. Grier, Phys. Rev. Lett. 109, 163903 (2012).
[CrossRef]

S. H. Lee, Y. Roichman, D. G. Grier, Opt. Express 18, 6988 (2010).
[CrossRef]

D. G. Grier, Nature 424, 810 (2003).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley and Sons, 1983).

Johnson, P. B.

P. B. Johnson, R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Juan, M. L.

M. L. Juan, M. Righini, R. Quidant, Nat. Photonics 5, 349 (2011).
[CrossRef]

Karásek, V.

O. Brzobohatý, V. Karásek, M. Šiler, L. Chvátal, T. Čižmár, P. Zemánek, Nat. Photonics 7, 123 (2013).
[CrossRef]

Kivshar, Y. S.

W. Liu, A. E. Miroshnichenko, D. N. Neshev, Y. S. Kivshar, ACS Nano 6, 5489 (2012).
[CrossRef]

Lavrinenko, A.

A. Novitsky, C. W. Qiu, A. Lavrinenko, Phys. Rev. Lett. 109, 023902 (2012).
[CrossRef]

Lee, S. H.

Lin, Z. F.

N. Wang, J. Chen, S. Y. Liu, Z. F. Lin, Phys. Rev. A 87, 063812 (2013).
[CrossRef]

J. Chen, J. Ng, Z. F. Lin, C. T. Chan, Nat. Photonics 5, 531 (2011).
[CrossRef]

J. Chen, J. Ng, P. Wang, Z. F. Lin, Opt. Lett. 35, 1674 (2010).

J. Ng, Z. F. Lin, C. T. Chan, Phys. Rev. Lett. 104, 103601 (2010).
[CrossRef]

J. Ng, Z. F. Lin, C. T. Chan, P. Sheng, Phys. Rev. B 72, 085130 (2005).
[CrossRef]

Liu, S. Y.

N. Wang, J. Chen, S. Y. Liu, Z. F. Lin, Phys. Rev. A 87, 063812 (2013).
[CrossRef]

Liu, W.

W. Liu, A. E. Miroshnichenko, D. N. Neshev, Y. S. Kivshar, ACS Nano 6, 5489 (2012).
[CrossRef]

Lock, J. A.

Marques, R.

R. P. Dominguez, F. L. Tejeira, R. Marques, J. A. S. Gil, New J. Phys. 13, 123017 (2011).
[CrossRef]

Marston, P. L.

P. L. Marston, J. Acoust. Soc. Am. 122, 3162 (2007).
[CrossRef]

P. L. Marston, J. Acoust. Soc. Am. 120, 3518 (2006).
[CrossRef]

Miroshnichenko, A. E.

W. Liu, A. E. Miroshnichenko, D. N. Neshev, Y. S. Kivshar, ACS Nano 6, 5489 (2012).
[CrossRef]

Mizrahi, A.

Mojahedi, M.

M. S. Wheeler, J. S. Aitchison, M. Mojahedi, Phys. Rev. B 73, 045105 (2006).
[CrossRef]

M. S. Wheeler, J. S. Aitchison, M. Mojahedi, Phys. Rev. B 72, 193103 (2005).
[CrossRef]

Neshev, D. N.

W. Liu, A. E. Miroshnichenko, D. N. Neshev, Y. S. Kivshar, ACS Nano 6, 5489 (2012).
[CrossRef]

Ng, J.

J. Chen, J. Ng, Z. F. Lin, C. T. Chan, Nat. Photonics 5, 531 (2011).
[CrossRef]

J. Ng, Z. F. Lin, C. T. Chan, Phys. Rev. Lett. 104, 103601 (2010).
[CrossRef]

J. Chen, J. Ng, P. Wang, Z. F. Lin, Opt. Lett. 35, 1674 (2010).

J. Ng, Z. F. Lin, C. T. Chan, P. Sheng, Phys. Rev. B 72, 085130 (2005).
[CrossRef]

Novitsky, A.

A. Novitsky, C. W. Qiu, A. Lavrinenko, Phys. Rev. Lett. 109, 023902 (2012).
[CrossRef]

A. Novitsky, C. W. Qiu, H. Wang, Phys. Rev. Lett. 107, 203601 (2011).
[CrossRef]

Olivik, M.

Z. Bouchal, M. Olivik, J. Mod. Opt. 42, 1555 (1995).
[CrossRef]

Padgett, M.

M. Padgett, R. Bowman, Nat. Photonics 5, 343 (2011).
[CrossRef]

Perkins, T. T.

T. T. Perkins, D. E. Smith, S. Chu, Science 264, 819 (1994).
[CrossRef]

Qiu, C. W.

A. Novitsky, C. W. Qiu, A. Lavrinenko, Phys. Rev. Lett. 109, 023902 (2012).
[CrossRef]

A. Novitsky, C. W. Qiu, H. Wang, Phys. Rev. Lett. 107, 203601 (2011).
[CrossRef]

Quidant, R.

M. L. Juan, M. Righini, R. Quidant, Nat. Photonics 5, 349 (2011).
[CrossRef]

Righini, M.

M. L. Juan, M. Righini, R. Quidant, Nat. Photonics 5, 349 (2011).
[CrossRef]

Roichman, Y.

Ruffner, D. B.

D. B. Ruffner, D. G. Grier, Phys. Rev. Lett. 109, 163903 (2012).
[CrossRef]

Saenz, J. J.

A. Dogariu, S. Sukhov, J. J. Saenz, Nat. Photonics 7, 24 (2013).
[CrossRef]

Sáenz, J.

J. Sáenz, Nat. Photonics 5, 514 (2011).
[CrossRef]

Sheng, P.

J. Ng, Z. F. Lin, C. T. Chan, P. Sheng, Phys. Rev. B 72, 085130 (2005).
[CrossRef]

Šiler, M.

O. Brzobohatý, V. Karásek, M. Šiler, L. Chvátal, T. Čižmár, P. Zemánek, Nat. Photonics 7, 123 (2013).
[CrossRef]

Smith, D. E.

T. T. Perkins, D. E. Smith, S. Chu, Science 264, 819 (1994).
[CrossRef]

Sukhov, S.

A. Dogariu, S. Sukhov, J. J. Saenz, Nat. Photonics 7, 24 (2013).
[CrossRef]

S. Sukhov, A. Dogariu, Opt. Lett. 35, 3847 (2010).
[CrossRef]

Tejeira, F. L.

R. P. Dominguez, F. L. Tejeira, R. Marques, J. A. S. Gil, New J. Phys. 13, 123017 (2011).
[CrossRef]

Turunen, J.

J. Turunen, A. T. Friberg, in Progress in Optics (Elsevier, 2009), Vol. 54, Chap. 1.

Wang, H.

A. Novitsky, C. W. Qiu, H. Wang, Phys. Rev. Lett. 107, 203601 (2011).
[CrossRef]

Wang, N.

N. Wang, J. Chen, S. Y. Liu, Z. F. Lin, Phys. Rev. A 87, 063812 (2013).
[CrossRef]

Wang, P.

Wheeler, M. S.

M. S. Wheeler, J. S. Aitchison, M. Mojahedi, Phys. Rev. B 73, 045105 (2006).
[CrossRef]

M. S. Wheeler, J. S. Aitchison, M. Mojahedi, Phys. Rev. B 72, 193103 (2005).
[CrossRef]

Zemánek, P.

O. Brzobohatý, V. Karásek, M. Šiler, L. Chvátal, T. Čižmár, P. Zemánek, Nat. Photonics 7, 123 (2013).
[CrossRef]

ACS Nano (1)

W. Liu, A. E. Miroshnichenko, D. N. Neshev, Y. S. Kivshar, ACS Nano 6, 5489 (2012).
[CrossRef]

Appl. Opt. (1)

J. Acoust. Soc. Am. (2)

P. L. Marston, J. Acoust. Soc. Am. 120, 3518 (2006).
[CrossRef]

P. L. Marston, J. Acoust. Soc. Am. 122, 3162 (2007).
[CrossRef]

J. Mod. Opt. (1)

Z. Bouchal, M. Olivik, J. Mod. Opt. 42, 1555 (1995).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

G. Gouesbet, J. Quant. Spectrosc. Radiat. Transfer 110, 1223 (2009).
[CrossRef]

Nat. Photonics (8)

F. M. Fazal, S. M. Block, Nat. Photonics 5, 318 (2011).
[CrossRef]

K. Dholakia, T. Čižmár, Nat. Photonics 5, 335 (2011).
[CrossRef]

M. Padgett, R. Bowman, Nat. Photonics 5, 343 (2011).
[CrossRef]

M. L. Juan, M. Righini, R. Quidant, Nat. Photonics 5, 349 (2011).
[CrossRef]

J. Sáenz, Nat. Photonics 5, 514 (2011).
[CrossRef]

A. Dogariu, S. Sukhov, J. J. Saenz, Nat. Photonics 7, 24 (2013).
[CrossRef]

O. Brzobohatý, V. Karásek, M. Šiler, L. Chvátal, T. Čižmár, P. Zemánek, Nat. Photonics 7, 123 (2013).
[CrossRef]

J. Chen, J. Ng, Z. F. Lin, C. T. Chan, Nat. Photonics 5, 531 (2011).
[CrossRef]

Nature (1)

D. G. Grier, Nature 424, 810 (2003).
[CrossRef]

New J. Phys. (1)

R. P. Dominguez, F. L. Tejeira, R. Marques, J. A. S. Gil, New J. Phys. 13, 123017 (2011).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. A (1)

N. Wang, J. Chen, S. Y. Liu, Z. F. Lin, Phys. Rev. A 87, 063812 (2013).
[CrossRef]

Phys. Rev. B (4)

J. Ng, Z. F. Lin, C. T. Chan, P. Sheng, Phys. Rev. B 72, 085130 (2005).
[CrossRef]

M. S. Wheeler, J. S. Aitchison, M. Mojahedi, Phys. Rev. B 72, 193103 (2005).
[CrossRef]

M. S. Wheeler, J. S. Aitchison, M. Mojahedi, Phys. Rev. B 73, 045105 (2006).
[CrossRef]

P. B. Johnson, R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Phys. Rev. Lett. (6)

J. Ng, Z. F. Lin, C. T. Chan, Phys. Rev. Lett. 104, 103601 (2010).
[CrossRef]

A. Ashkin, Phys. Rev. Lett. 24, 156 (1970).
[CrossRef]

A. Novitsky, C. W. Qiu, H. Wang, Phys. Rev. Lett. 107, 203601 (2011).
[CrossRef]

S. Aukhov, A. Dogariu, Phys. Rev. Lett. 107, 203602 (2011).
[CrossRef]

A. Novitsky, C. W. Qiu, A. Lavrinenko, Phys. Rev. Lett. 109, 023902 (2012).
[CrossRef]

D. B. Ruffner, D. G. Grier, Phys. Rev. Lett. 109, 163903 (2012).
[CrossRef]

Science (1)

T. T. Perkins, D. E. Smith, S. Chu, Science 264, 819 (1994).
[CrossRef]

Other (3)

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley and Sons, 1983).

G. Gouesbet, G. Grehan, Generalized Lorenz–Mie Theories (Springer, 2011).

J. Turunen, A. T. Friberg, in Progress in Optics (Elsevier, 2009), Vol. 54, Chap. 1.

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Figures (4)

Fig. 1.
Fig. 1.

Mie coefficients a 1 and b 1 of an Ag at Si core–shell nanoparticle versus the wavelength λ , showing that the electric and magnetic dipolar resonances can be tuned to overlap spectrally. Higher order modes are negligibly small and are thus not shown.

Fig. 2.
Fig. 2.

Polar plots of the normalized scattering intensity S ( θ ) for different kinds of Bessel beams with the cone angle α = 70 ° and the wavelength λ = 1294 nm . (a)  l = 0 , c 2 = i c 1 = i / 2 , (b)  l = 1 , c 1 = 1 , c 2 = 0 , (c)  l = 1 , c 1 = 0 , c 2 = i , and (d)  l = 1 , c 2 = i c 1 = i / 2 .

Fig. 3.
Fig. 3.

Longitudinal optical force f z as a function of wavelength λ for different kinds of Bessel beams. The cone angle of the Bessel beams is α = 70 ° and E 0 = 10 6 V / m .

Fig. 4.
Fig. 4.

(a)  f z (red line) and k 11 (blue dotted line) as a function of wavelength λ for a first-order Bessel beam with mixed polarization ( α = 70 ° , c 2 = i c 1 = i / 2 ) and E 0 = 10 6 V / m . The red solid line denotes a region of stable transverse trapping (i.e., k 11 < 0 ), while the red dashed line represents a region of unstable transverse trapping (i.e., k 11 > 0 ). (b) The trajectory of a core–shell nanoparticle that is initially located at k x = 0 , k y = 0.8 with zero velocity in the presence of ambient damping. The damping constant is 17.7 pN μs/μm which is 1.2 times that of the critical damping constant [35,36]. The incident wavelength is λ = 1300 nm , corresponding to the black square marked in (a).

Equations (8)

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E = E 0 ( i c 1 M l + c 2 N l ) e i ω t ,
M l = 1 q [ i l q k ρ J l ( q k ρ ) e ρ J l ( q k ρ ) e ϕ ] e i l ϕ + i β k z , N l = i β e z × M l + J l ( q k ρ ) e z e i l ϕ + i β k z ,
f z = ( W ext cos α W sca cos θ ) / c ,
S ( θ , ϕ ) = lim k r ( k r ) 2 | E s ( r , θ , ϕ ) | 2 | E 0 | 2 ,
S ( θ ) = 9 q 2 ( | a 1 c 2 | 2 + | b 1 c 1 | 2 ) sin 2 θ / ( 4 k 2 ) ,
S ( θ ) = 9 [ | a 1 ( i c 2 β c 1 ) cos θ + b 1 ( i c 2 c 1 β ) | 2 + | a 1 ( i c 2 β c 1 ) + b 1 ( i c 2 c 1 β ) cos θ | 2 ] / 16 ,
S ( θ ) = 9 | a 1 | 2 ( cos θ + 1 ) 2 ( 1 + β ) 2 / 32 ,
K ^ = [ k 11 k 12 k 12 k 11 ] ,

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