Abstract

The extended boundary condition method, also called the null-field method, provides a semianalytic solution to the problem of electromagnetic scattering by a particle by constructing a transition matrix (T-matrix) that links the scattered field to the incident field. This approach requires the computation of specific integrals over the particle surface, which are typically evaluated numerically. We introduce here a new set of simplified expressions for these integrals in the commonly studied case of axisymmetric particles. Simplifications are obtained using the differentiation properties of the radial functions (spherical Bessel) and angular functions (associated Legendre functions) and integrations by parts. The resulting simplified expressions not only lead to faster computations, but also reduce the risks of loss of precision and provide a simpler framework for further analytical work.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. C. Waterman, Phys. Rev. D 3, 825 (1971).
    [CrossRef]
  2. P. Barber and C. Yeh, Appl. Opt. 14, 2864 (1975).
    [PubMed]
  3. P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
    [CrossRef]
  4. L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves (Wiley, 2000).
    [CrossRef]
  5. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles, 3rd ed. (Cambridge University, 2002).
  6. A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles: Null-field Method with Discrete Sources: Theory and Programs, Springer Series in Optical Sciences (Springer, 2006), Vol.  124.
  7. P. C. Waterman, Proc. IEEE 53, 805 (1965).
    [CrossRef]
  8. M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, J. Quant. Spectrosc. Radiat. Transfer 88, 357 (2004).
  9. J. B. Schneider and I. C. Peden, IEEE Trans. Antennas Propag. 36, 1317 (1988).
    [CrossRef]
  10. F. Kahnert, J. Stamnes, and K. Stamnes, Appl. Opt. 40, 3110 (2001).
    [CrossRef]
  11. R. Boyack and E. C. Le Ru, Phys. Chem. Chem. Phys. 11, 7398 (2009).
    [CrossRef] [PubMed]
  12. B. N. Khlebtsov and N. G. Khlebtsov, J. Phys. Chem. C 111, 11516 (2007).
    [CrossRef]
  13. P. W. Barber, R. K. Chang, and H. Massoudi, Phys. Rev. B 27, 7251 (1983).
    [CrossRef]
  14. P. W. Barber, R. K. Chang, and H. Massoudi, Phys. Rev. Lett. 50, 997 (1983).
    [CrossRef]
  15. P. C. Waterman, J. Acoust. Soc. Am. 45, 1417 (1969).
    [CrossRef]
  16. B. Peterson and S. Ström, Phys. Rev. D 10, 2670 (1974).
    [CrossRef]
  17. B. Peterson and S. Ström, Phys. Rev. D 8, 3661 (1973).
    [CrossRef]
  18. A. Doicu and T. Wriedt, Optics Commun. 139, 85 (1997).
    [CrossRef]
  19. A. Doicu and T. Wriedt, J. Quant. Spectrosc. Radiat. Transfer 111, 466 (2010).
    [CrossRef]
  20. F. Xu, J. A. Lock, and G. Gouesbet, Phys. Rev. A 81, 043824 (2010).
    [CrossRef]
  21. D. Petrov, Y. Shkuratov, and G. Videen, Opt. Lett. 32, 1168 (2007).
    [CrossRef] [PubMed]
  22. M. Mishchenko and L. Travis, Optics. Commun. 109, 16 (1994).
    [CrossRef]
  23. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, 1972).

2010

A. Doicu and T. Wriedt, J. Quant. Spectrosc. Radiat. Transfer 111, 466 (2010).
[CrossRef]

F. Xu, J. A. Lock, and G. Gouesbet, Phys. Rev. A 81, 043824 (2010).
[CrossRef]

2009

R. Boyack and E. C. Le Ru, Phys. Chem. Chem. Phys. 11, 7398 (2009).
[CrossRef] [PubMed]

2007

B. N. Khlebtsov and N. G. Khlebtsov, J. Phys. Chem. C 111, 11516 (2007).
[CrossRef]

D. Petrov, Y. Shkuratov, and G. Videen, Opt. Lett. 32, 1168 (2007).
[CrossRef] [PubMed]

2004

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, J. Quant. Spectrosc. Radiat. Transfer 88, 357 (2004).

2001

1997

A. Doicu and T. Wriedt, Optics Commun. 139, 85 (1997).
[CrossRef]

1994

M. Mishchenko and L. Travis, Optics. Commun. 109, 16 (1994).
[CrossRef]

1988

J. B. Schneider and I. C. Peden, IEEE Trans. Antennas Propag. 36, 1317 (1988).
[CrossRef]

1983

P. W. Barber, R. K. Chang, and H. Massoudi, Phys. Rev. B 27, 7251 (1983).
[CrossRef]

P. W. Barber, R. K. Chang, and H. Massoudi, Phys. Rev. Lett. 50, 997 (1983).
[CrossRef]

1975

1974

B. Peterson and S. Ström, Phys. Rev. D 10, 2670 (1974).
[CrossRef]

1973

B. Peterson and S. Ström, Phys. Rev. D 8, 3661 (1973).
[CrossRef]

1971

P. C. Waterman, Phys. Rev. D 3, 825 (1971).
[CrossRef]

1969

P. C. Waterman, J. Acoust. Soc. Am. 45, 1417 (1969).
[CrossRef]

1965

P. C. Waterman, Proc. IEEE 53, 805 (1965).
[CrossRef]

Abramowitz, M.

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

Babenko, V. A.

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, J. Quant. Spectrosc. Radiat. Transfer 88, 357 (2004).

Barber, P.

Barber, P. W.

P. W. Barber, R. K. Chang, and H. Massoudi, Phys. Rev. Lett. 50, 997 (1983).
[CrossRef]

P. W. Barber, R. K. Chang, and H. Massoudi, Phys. Rev. B 27, 7251 (1983).
[CrossRef]

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

Boyack, R.

R. Boyack and E. C. Le Ru, Phys. Chem. Chem. Phys. 11, 7398 (2009).
[CrossRef] [PubMed]

Chang, R. K.

P. W. Barber, R. K. Chang, and H. Massoudi, Phys. Rev. B 27, 7251 (1983).
[CrossRef]

P. W. Barber, R. K. Chang, and H. Massoudi, Phys. Rev. Lett. 50, 997 (1983).
[CrossRef]

Ding, K.-H.

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves (Wiley, 2000).
[CrossRef]

Doicu, A.

A. Doicu and T. Wriedt, J. Quant. Spectrosc. Radiat. Transfer 111, 466 (2010).
[CrossRef]

A. Doicu and T. Wriedt, Optics Commun. 139, 85 (1997).
[CrossRef]

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles: Null-field Method with Discrete Sources: Theory and Programs, Springer Series in Optical Sciences (Springer, 2006), Vol.  124.

Eremin, Y. A.

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles: Null-field Method with Discrete Sources: Theory and Programs, Springer Series in Optical Sciences (Springer, 2006), Vol.  124.

Gouesbet, G.

F. Xu, J. A. Lock, and G. Gouesbet, Phys. Rev. A 81, 043824 (2010).
[CrossRef]

Hill, S. C.

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

Kahnert, F.

Khlebtsov, B. N.

B. N. Khlebtsov and N. G. Khlebtsov, J. Phys. Chem. C 111, 11516 (2007).
[CrossRef]

Khlebtsov, N. G.

B. N. Khlebtsov and N. G. Khlebtsov, J. Phys. Chem. C 111, 11516 (2007).
[CrossRef]

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, J. Quant. Spectrosc. Radiat. Transfer 88, 357 (2004).

Kong, J. A.

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves (Wiley, 2000).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles, 3rd ed. (Cambridge University, 2002).

Le Ru, E. C.

R. Boyack and E. C. Le Ru, Phys. Chem. Chem. Phys. 11, 7398 (2009).
[CrossRef] [PubMed]

Lock, J. A.

F. Xu, J. A. Lock, and G. Gouesbet, Phys. Rev. A 81, 043824 (2010).
[CrossRef]

Massoudi, H.

P. W. Barber, R. K. Chang, and H. Massoudi, Phys. Rev. Lett. 50, 997 (1983).
[CrossRef]

P. W. Barber, R. K. Chang, and H. Massoudi, Phys. Rev. B 27, 7251 (1983).
[CrossRef]

Mishchenko, M.

M. Mishchenko and L. Travis, Optics. Commun. 109, 16 (1994).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, J. Quant. Spectrosc. Radiat. Transfer 88, 357 (2004).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles, 3rd ed. (Cambridge University, 2002).

Peden, I. C.

J. B. Schneider and I. C. Peden, IEEE Trans. Antennas Propag. 36, 1317 (1988).
[CrossRef]

Peterson, B.

B. Peterson and S. Ström, Phys. Rev. D 10, 2670 (1974).
[CrossRef]

B. Peterson and S. Ström, Phys. Rev. D 8, 3661 (1973).
[CrossRef]

Petrov, D.

Schneider, J. B.

J. B. Schneider and I. C. Peden, IEEE Trans. Antennas Propag. 36, 1317 (1988).
[CrossRef]

Shkuratov, Y.

Stamnes, J.

Stamnes, K.

Stegun, I. A.

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

Ström, S.

B. Peterson and S. Ström, Phys. Rev. D 10, 2670 (1974).
[CrossRef]

B. Peterson and S. Ström, Phys. Rev. D 8, 3661 (1973).
[CrossRef]

Travis, L.

M. Mishchenko and L. Travis, Optics. Commun. 109, 16 (1994).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles, 3rd ed. (Cambridge University, 2002).

Tsang, L.

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves (Wiley, 2000).
[CrossRef]

Videen, G.

D. Petrov, Y. Shkuratov, and G. Videen, Opt. Lett. 32, 1168 (2007).
[CrossRef] [PubMed]

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, J. Quant. Spectrosc. Radiat. Transfer 88, 357 (2004).

Waterman, P. C.

P. C. Waterman, Phys. Rev. D 3, 825 (1971).
[CrossRef]

P. C. Waterman, J. Acoust. Soc. Am. 45, 1417 (1969).
[CrossRef]

P. C. Waterman, Proc. IEEE 53, 805 (1965).
[CrossRef]

Wriedt, T.

A. Doicu and T. Wriedt, J. Quant. Spectrosc. Radiat. Transfer 111, 466 (2010).
[CrossRef]

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, J. Quant. Spectrosc. Radiat. Transfer 88, 357 (2004).

A. Doicu and T. Wriedt, Optics Commun. 139, 85 (1997).
[CrossRef]

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles: Null-field Method with Discrete Sources: Theory and Programs, Springer Series in Optical Sciences (Springer, 2006), Vol.  124.

Xu, F.

F. Xu, J. A. Lock, and G. Gouesbet, Phys. Rev. A 81, 043824 (2010).
[CrossRef]

Yeh, C.

Appl. Opt.

IEEE Trans. Antennas Propag.

J. B. Schneider and I. C. Peden, IEEE Trans. Antennas Propag. 36, 1317 (1988).
[CrossRef]

J. Acoust. Soc. Am.

P. C. Waterman, J. Acoust. Soc. Am. 45, 1417 (1969).
[CrossRef]

J. Phys. Chem. C

B. N. Khlebtsov and N. G. Khlebtsov, J. Phys. Chem. C 111, 11516 (2007).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

A. Doicu and T. Wriedt, J. Quant. Spectrosc. Radiat. Transfer 111, 466 (2010).
[CrossRef]

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, J. Quant. Spectrosc. Radiat. Transfer 88, 357 (2004).

Opt. Lett.

Optics Commun.

A. Doicu and T. Wriedt, Optics Commun. 139, 85 (1997).
[CrossRef]

Optics. Commun.

M. Mishchenko and L. Travis, Optics. Commun. 109, 16 (1994).
[CrossRef]

Phys. Chem. Chem. Phys.

R. Boyack and E. C. Le Ru, Phys. Chem. Chem. Phys. 11, 7398 (2009).
[CrossRef] [PubMed]

Phys. Rev. A

F. Xu, J. A. Lock, and G. Gouesbet, Phys. Rev. A 81, 043824 (2010).
[CrossRef]

Phys. Rev. B

P. W. Barber, R. K. Chang, and H. Massoudi, Phys. Rev. B 27, 7251 (1983).
[CrossRef]

Phys. Rev. D

B. Peterson and S. Ström, Phys. Rev. D 10, 2670 (1974).
[CrossRef]

B. Peterson and S. Ström, Phys. Rev. D 8, 3661 (1973).
[CrossRef]

P. C. Waterman, Phys. Rev. D 3, 825 (1971).
[CrossRef]

Phys. Rev. Lett.

P. W. Barber, R. K. Chang, and H. Massoudi, Phys. Rev. Lett. 50, 997 (1983).
[CrossRef]

Proc. IEEE

P. C. Waterman, Proc. IEEE 53, 805 (1965).
[CrossRef]

Other

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves (Wiley, 2000).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles, 3rd ed. (Cambridge University, 2002).

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles: Null-field Method with Discrete Sources: Theory and Programs, Springer Series in Optical Sciences (Springer, 2006), Vol.  124.

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Equations (28)

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

Q = Q 11 Q 12 Q 21 Q 22 = i k 1 2 s J 21 + J 12 s J 11 + J 22 s J 22 + J 11 s J 12 + J 21 ,
i s k 1 2 J n k 11 A n A k = 0 π d θ sin θ ξ n ψ k [ π n τ k + τ n π k ] ,
i s k 1 2 J n k 22 A n A k = 0 π d θ sin θ { ξ n ψ k [ π n τ k + τ n π k ] + [ n ( n + 1 ) ξ n ψ k + k ( k + 1 ) s ξ n ψ k ] x θ x 2 π n d k } ,
s k 1 2 J n k 12 A n A k = 0 π d θ sin θ ψ k { ξ n [ π n π k + τ n τ k ] + x θ x 2 n ( n + 1 ) ξ n d n τ k } ,
s k 1 2 J n k 21 A n A k = 0 π d θ sin θ ξ n { ψ k [ π n π k + τ n τ k ] + x θ s x 2 k ( k + 1 ) ψ k τ n d k } ,
ξ n ξ n ( x ( θ ) ) and ξ n ξ n ( x ( θ ) ) , ψ n ψ n ( s x ( θ ) ) and ψ n ψ n ( s x ( θ ) ) ,
π n π m n ( θ ) , τ n τ m n ( θ ) , and d n d 0 m n ( θ ) ,
[ π n τ k + τ n π k ] sin θ = m ( d n d k ) ,
m d n ( θ = 0 ) = m d n ( θ = π ) = 0 ,
[ π n π k + τ n τ k ] sin θ = ( τ k d n sin θ ) + k ( k + 1 ) d n d k sin θ ,
[ π n π k + τ n τ k ] sin θ = ( τ n d k sin θ ) + n ( n + 1 ) d n d k sin θ .
ξ n ( x ) = [ n ( n + 1 ) x 2 1 ] ξ n ( x ) ,
[ n ( n + 1 ) s 2 k ( k + 1 ) ] x θ ξ n ψ k = d d θ [ k ( k + 1 ) ξ n ψ k n ( n + 1 ) s ξ n ψ k ] + [ n ( n + 1 ) k ( k + 1 ) ] s x θ ξ n ψ k .
i s k 1 2 J n k 11 A n A k = ( K n k 2 s K n k 1 ) ,
K n k 1 = 0 π d θ m d n d k x θ ξ n ψ k ,
K n k 2 = 0 π d θ m d n d k x θ ξ n ψ k .
i s k 1 2 J n k 22 A n A k = ( s K n k 2 + K n k 1 ) .
Q n k 12 = A n A k s 2 1 s K n k 1 ,
Q n k 21 = A n A k 1 s 2 s K n k 2 .
s k 1 2 J n k 21 A n A k = ( s L n k 1 + L n k 3 ) ,
s k 1 2 J n k 12 A n A k = ( L n k 2 L n k 4 ) ,
L n k 1 = 0 π d θ sin θ x θ τ n d k ξ n ψ k ,
L n k 2 = 0 π d θ sin θ x θ d n τ k ξ n ψ k ,
L n k 3 = 0 π d θ sin θ d k ψ k [ x θ τ n ξ n n ( n + 1 ) d n ξ n ] ,
L n k 4 = 0 π d θ sin θ d n ξ n [ s x θ τ k ψ k k ( k + 1 ) d k ψ k ] .
s L n k 3 L n k 4 = s 2 n ( n + 1 ) k ( k + 1 ) n ( n + 1 ) k ( k + 1 ) [ L n k 1 L n k 2 ] .
Q n k 11 = i A n A k ( s 2 1 ) / s n ( n + 1 ) k ( k + 1 ) [ n ( n + 1 ) L n k 2 k ( k + 1 ) L n k 1 ] ,
Q n k 22 = i A n A k ( s 2 1 ) / s [ L n k 3 + s n ( n + 1 ) ( L n k 2 L n k 1 ) n ( n + 1 ) k ( k + 1 ) ] .

Metrics