Abstract

Multipole expansions of Bessel and Gaussian beams, suitable for use in Mie scattering calculations, are derived. These results allow Mie scattering calculations to be carried out considerably faster than existing methods, something that is of particular interest for time evolution simulations where large numbers of scattering calculations must be performed. An analytic result is derived for the Bessel beam that improves on a previously published expression requiring the evaluation of an integral. An analogous expression containing a single integral, similar to existing results quoted, but not derived, in literature, is derived for a Gaussian beam, valid from the paraxial limit all the way to arbitrarily high numerical apertures.

© 2009 Optical Society of America

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References

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  1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  2. J. P. Barton, W. Ma, S. A. Schaub, and D. R. Alexander, “Electromagnetic field for a beam incident on two adjacent spherical particles,” Appl. Opt. 30, 4706-4715 (1991).
    [CrossRef] [PubMed]
  3. Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573-4588 (1995).
    [CrossRef] [PubMed]
  4. D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London, Ser. A 433, 599-614 (1991).
    [CrossRef]
  5. J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
    [CrossRef]
  6. 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]
  7. M. Kawano, J. T. Blakely, R. Gordon, and D. Sinton, “Theory of dielectric microsphere dynamics in a dual-beam optical trap,” Opt. Express 16, 9306-9317 (2008).
    [CrossRef] [PubMed]
  8. T. Čižmár, V. Kollárová, Z. Bouchal, and P. Zemánek, “Sub-micron particle organization by self-imaging of non-diffracting beams,” New J. Phys. 8, 43 (2006).
    [CrossRef]
  9. J. M. Taylor, L. Y. Wong, C. D. Bain, and G. D. Love, “Emergent properties in optically bound matter,” Opt. Express 16, 6921-6929 (2008).
    [CrossRef] [PubMed]
  10. 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]
  11. G. Gouesbet, G. Grehan, and B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A 7, 998-1007 (1990).
    [CrossRef]
  12. P. C. Chaumet, “Fully vectorial highly nonparaxial beam close to the waist,” J. Opt. Soc. Am. A 23, 3197-3202 (2006).
    [CrossRef]
  13. A. D. Ward, M. G. Berry, C. D. Mellor, and C. D. Bain, “Optical sculpture: controlled deformation of emulsion droplets with ultralow interfacial tensions using optical tweezers,” Chem. Commun. (Cambridge) 2006, 4515-4517 (2006).
    [CrossRef]
  14. P. A. Maia Neto and H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. 50, 702-708 (2000).
    [CrossRef]
  15. A. Mazolli, P. A. Maia Neto, and H. M. Nussenzveig, “Theory of trapping forces in optical tweezers,” Proc. R. Soc. London, Ser. A 459, 3021-3041 (2003).
    [CrossRef]
  16. D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851-2861 (1994).
    [CrossRef]
  17. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).
  18. A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. 136, 114-124 (1996).
    [CrossRef]
  19. J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800-2802 (1989).
    [CrossRef]
  20. Y. I. Salamin, “Fields of a Gaussian beam beyond the paraxial approximation,” Appl. Phys. B 86, 319-326 (2007).
    [CrossRef]
  21. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
    [CrossRef]
  22. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505-1509 (1980).
    [CrossRef] [PubMed]

2008 (2)

2007 (1)

Y. I. Salamin, “Fields of a Gaussian beam beyond the paraxial approximation,” Appl. Phys. B 86, 319-326 (2007).
[CrossRef]

2006 (3)

P. C. Chaumet, “Fully vectorial highly nonparaxial beam close to the waist,” J. Opt. Soc. Am. A 23, 3197-3202 (2006).
[CrossRef]

A. D. Ward, M. G. Berry, C. D. Mellor, and C. D. Bain, “Optical sculpture: controlled deformation of emulsion droplets with ultralow interfacial tensions using optical tweezers,” Chem. Commun. (Cambridge) 2006, 4515-4517 (2006).
[CrossRef]

T. Čižmár, V. Kollárová, Z. Bouchal, and P. Zemánek, “Sub-micron particle organization by self-imaging of non-diffracting beams,” New J. Phys. 8, 43 (2006).
[CrossRef]

2005 (1)

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[CrossRef]

2003 (2)

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]

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

2000 (1)

P. A. Maia Neto and H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. 50, 702-708 (2000).
[CrossRef]

1996 (1)

A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. 136, 114-124 (1996).
[CrossRef]

1995 (1)

1994 (1)

1991 (2)

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London, Ser. A 433, 599-614 (1991).
[CrossRef]

J. P. Barton, W. Ma, S. A. Schaub, and D. R. Alexander, “Electromagnetic field for a beam incident on two adjacent spherical particles,” Appl. Opt. 30, 4706-4715 (1991).
[CrossRef] [PubMed]

1990 (1)

1989 (2)

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. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

1980 (1)

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Alexander, D. R.

J. P. Barton, W. Ma, S. A. Schaub, and D. R. Alexander, “Electromagnetic field for a beam incident on two adjacent spherical particles,” Appl. Opt. 30, 4706-4715 (1991).
[CrossRef] [PubMed]

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

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]

Bain, C. D.

J. M. Taylor, L. Y. Wong, C. D. Bain, and G. D. Love, “Emergent properties in optically bound matter,” Opt. Express 16, 6921-6929 (2008).
[CrossRef] [PubMed]

A. D. Ward, M. G. Berry, C. D. Mellor, and C. D. Bain, “Optical sculpture: controlled deformation of emulsion droplets with ultralow interfacial tensions using optical tweezers,” Chem. Commun. (Cambridge) 2006, 4515-4517 (2006).
[CrossRef]

Barton, J. P.

J. P. Barton, W. Ma, S. A. Schaub, and D. R. Alexander, “Electromagnetic field for a beam incident on two adjacent spherical particles,” Appl. Opt. 30, 4706-4715 (1991).
[CrossRef] [PubMed]

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

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]

Berry, M. G.

A. D. Ward, M. G. Berry, C. D. Mellor, and C. D. Bain, “Optical sculpture: controlled deformation of emulsion droplets with ultralow interfacial tensions using optical tweezers,” Chem. Commun. (Cambridge) 2006, 4515-4517 (2006).
[CrossRef]

Blakely, J. T.

Bohren, C. F.

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

Bouchal, Z.

T. Čižmár, V. Kollárová, Z. Bouchal, and P. Zemánek, “Sub-micron particle organization by self-imaging of non-diffracting beams,” New J. Phys. 8, 43 (2006).
[CrossRef]

Chan, C. T.

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[CrossRef]

Chaumet, P. C.

Cižmár, T.

T. Čižmár, V. Kollárová, Z. Bouchal, and P. Zemánek, “Sub-micron particle organization by self-imaging of non-diffracting beams,” New J. Phys. 8, 43 (2006).
[CrossRef]

Doicu, A.

A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. 136, 114-124 (1996).
[CrossRef]

Gordon, R.

Gouesbet, G.

Grehan, G.

Heckenberg, N. R.

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]

Huffman, D. R.

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

Kawano, M.

Kollárová, V.

T. Čižmár, V. Kollárová, Z. Bouchal, and P. Zemánek, “Sub-micron particle organization by self-imaging of non-diffracting beams,” New J. Phys. 8, 43 (2006).
[CrossRef]

Lin, Z. F.

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[CrossRef]

Love, G. D.

Ma, W.

Mackowski, D. W.

D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851-2861 (1994).
[CrossRef]

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London, Ser. A 433, 599-614 (1991).
[CrossRef]

Maheu, B.

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, Ser. A 459, 3021-3041 (2003).
[CrossRef]

P. A. Maia Neto and H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. 50, 702-708 (2000).
[CrossRef]

Mazolli, A.

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

Mellor, C. D.

A. D. Ward, M. G. Berry, C. D. Mellor, and C. D. Bain, “Optical sculpture: controlled deformation of emulsion droplets with ultralow interfacial tensions using optical tweezers,” Chem. Commun. (Cambridge) 2006, 4515-4517 (2006).
[CrossRef]

Ng, J.

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[CrossRef]

Nieminen, T. A.

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]

Nussenzveig, H. M.

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

P. A. Maia Neto and H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. 50, 702-708 (2000).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Rubinsztein-Dunlop, H.

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]

Salamin, Y. I.

Y. I. Salamin, “Fields of a Gaussian beam beyond the paraxial approximation,” Appl. Phys. B 86, 319-326 (2007).
[CrossRef]

Schaub, S. A.

J. P. Barton, W. Ma, S. A. Schaub, and D. R. Alexander, “Electromagnetic field for a beam incident on two adjacent spherical particles,” Appl. Opt. 30, 4706-4715 (1991).
[CrossRef] [PubMed]

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]

Sheng, P.

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[CrossRef]

Sinton, D.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Taylor, J. M.

Ward, A. D.

A. D. Ward, M. G. Berry, C. D. Mellor, and C. D. Bain, “Optical sculpture: controlled deformation of emulsion droplets with ultralow interfacial tensions using optical tweezers,” Chem. Commun. (Cambridge) 2006, 4515-4517 (2006).
[CrossRef]

Wiscombe, W. J.

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Wong, L. Y.

Wriedt, T.

A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. 136, 114-124 (1996).
[CrossRef]

Xu, Y.-L.

Zemánek, P.

T. Čižmár, V. Kollárová, Z. Bouchal, and P. Zemánek, “Sub-micron particle organization by self-imaging of non-diffracting beams,” New J. Phys. 8, 43 (2006).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. B (1)

Y. I. Salamin, “Fields of a Gaussian beam beyond the paraxial approximation,” Appl. Phys. B 86, 319-326 (2007).
[CrossRef]

Chem. Commun. (Cambridge) (1)

A. D. Ward, M. G. Berry, C. D. Mellor, and C. D. Bain, “Optical sculpture: controlled deformation of emulsion droplets with ultralow interfacial tensions using optical tweezers,” Chem. Commun. (Cambridge) 2006, 4515-4517 (2006).
[CrossRef]

Europhys. Lett. (1)

P. A. Maia Neto and H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. 50, 702-708 (2000).
[CrossRef]

J. Appl. Phys. (2)

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. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

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

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

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]

New J. Phys. (1)

T. Čižmár, V. Kollárová, Z. Bouchal, and P. Zemánek, “Sub-micron particle organization by self-imaging of non-diffracting beams,” New J. Phys. 8, 43 (2006).
[CrossRef]

Opt. Commun. (1)

A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. 136, 114-124 (1996).
[CrossRef]

Opt. Express (2)

Phys. Rev. B (1)

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[CrossRef]

Proc. R. Soc. London, Ser. A (3)

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London, Ser. A 433, 599-614 (1991).
[CrossRef]

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

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Other (2)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

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

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Figures (2)

Fig. 1
Fig. 1

Execution time for alternative BSC calculation algorithms as a function of sphere radial position k r relative to the beam axis. For low NAs our single integral calculation is shown to be roughly ten times faster than a double integral calculation over the plane wave decomposition of the beam. For a Bessel beam our analytical result is shown to be up to ten times faster than a single integral calculation. For reference, execution times are also shown for force calculation and for a two-sphere multiple scattering calculation.

Fig. 2
Fig. 2

Execution time for high-NA BSC calculation algorithms as a function of beam NA. Our single integral calculation (labeled “Integral”) is shown to be over 100 times faster than solving the appropriate linear system of equations (labeled “Inversion”). For reference, execution times are also shown for force calculation and for a two-sphere multiple scattering calculation. The timings are independent of particle size.

Equations (15)

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E = i 2 n + 1 4 π ( n m ) ! ( n + m ) ! n = 1 m = n n ( p m n N m n ( 1 ) + q m n M m n ( 1 ) ) .
p m n = i 4 π 2 n + 1 ( n + m ) ! ( n m ) ! × 0 2 π 0 π E N m n * sin θ d θ d ϕ 0 2 π 0 π N m n 2 sin θ d θ d ϕ ,
A l m = i p m n 2 π ( k a ) , B l m = n ext q m n 2 π ( k a ) .
E ( r ) = E 0 0 2 π e 0 ( θ , ϕ ) e i k r d ϕ ,
{ p m n q m n } = U n ( e θ { τ ̃ m n π ̃ m n } i e ϕ { π ̃ m n τ ̃ m n } ) e i m ϕ e i k r 0 ,
U n = 4 π i n n ( n + 1 ) ,
π ̃ m n ( cos θ ) = 2 n + 1 4 π ( n m ) ! ( n + m ) ! m sin θ P n m ( cos θ ) ,
τ ̃ m n ( cos θ ) = 2 n + 1 4 π ( n m ) ! ( n + m ) ! d d θ P n m ( cos θ ) .
{ p m n q m n } = E 0 U n × 0 2 π [ cos ( ϕ ) { τ ̃ m n π ̃ m n } + i sin ( ϕ ) { π ̃ m n τ ̃ m n } ] e i m ϕ e i k r 0 d ϕ .
{ p m n q m n } = E 0 U n e i k z cos θ × [ { τ ̃ m n π ̃ m n } I + + { π ̃ m n τ ̃ m n } I ] ,
I ± = 1 2 0 2 π e i ( 1 m ) ϕ e i ρ cos [ ϕ + ϕ 0 + ( π 2 ) ] d ϕ ± 1 2 0 2 π e i ( 1 m ) ϕ e i ρ cos [ ϕ + ϕ 0 + ( π 2 ) ] d ϕ .
I ± = π ( e i ( m 1 ) ϕ 0 J 1 m ( ρ ) ± e i ( m + 1 ) ϕ 0 J 1 m ( ρ ) ) .
2 n + 1 4 π ( n m ) ! ( n + m ) ! { N m n M m n } = ( { τ ̃ m n i π ̃ m n } i θ + { i π ̃ m n τ ̃ m n } i ϕ ) e i m ϕ k R .
E = E 0 k 2 w 0 2 2 i k R cos θ e ( γ sin θ ) 2 e i k R e i r i r ( cos ϕ i θ sin ϕ i ϕ ) ,
{ p m n q m n } = U n 0 θ 0 E ( θ ) ( { τ ̃ m n π ̃ m n } I + + { π ̃ m n τ ̃ m n } I ) sin θ d θ ,

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