Abstract

The Mie scattering coefficients satisfy recurrence relations:an−1, bn−1, an, and bn determine an+1, and bn+1. It is therefore possible, in principle, to generate the entire set from the first four, which has a simple interpretation. Each term in a multipole expansion of an electrostatic field can be obtained by differentiating the preceding term. The Mie coefficients are terms in a multipole expansion of a particular electromagnetic field, namely, that scattered by an arbitrary sphere. By analogy, it is not surprising that all these coefficients can be generated from the electric and magnetic dipole and quadrupole terms. Moreover, the recurrence relations for the Mie coefficients contain finite differences, in analogy with the infinitesimal differences (derivatives) in the multipole expansion of an electrostatic field.

© 1987 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec. 9.25.
  2. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 9.
  3. M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), Sec. 13.5.
  4. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chaps. 3 and 4.
  5. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small ParticlesWiley-Interscience, New York, 1983), Chap. 4.
  6. J. V. Dave, Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere, IBM Order No. 360D-17.4.002 (IBM Scientific Center, Palo Alto, Calif., 1968).
  7. W. J. Lentz, “Generating Bessel functions in Mie scattering calculations using continued fractions,” Appl. Opt. 15, 668–671 (1976).
    [CrossRef] [PubMed]
  8. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  9. B. Verner, “Note on the recurrence between Mie’s coefficients,”J. Opt. Soc. Am. 66, 1424–1425 (1976).
    [CrossRef]
  10. J. C. Maxwell, A Treatise on Electricity and Magnetism (Dover, New York, 1954) Vol. 1, p. 196.
  11. E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Cambridge U. Press, Cambridge, 1931), pp. 129–132.
  12. W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1962), p. 15.

1980 (1)

1976 (2)

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small ParticlesWiley-Interscience, New York, 1983), Chap. 4.

Born, M.

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), Sec. 13.5.

Dave, J. V.

J. V. Dave, Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere, IBM Order No. 360D-17.4.002 (IBM Scientific Center, Palo Alto, Calif., 1968).

Hobson, E. W.

E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Cambridge U. Press, Cambridge, 1931), pp. 129–132.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small ParticlesWiley-Interscience, New York, 1983), Chap. 4.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chaps. 3 and 4.

Lentz, W. J.

Maxwell, J. C.

J. C. Maxwell, A Treatise on Electricity and Magnetism (Dover, New York, 1954) Vol. 1, p. 196.

Panofsky, W. K. H.

W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1962), p. 15.

Phillips, M.

W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1962), p. 15.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec. 9.25.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 9.

Verner, B.

Wiscombe, W. J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), Sec. 13.5.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Other (9)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec. 9.25.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 9.

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), Sec. 13.5.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chaps. 3 and 4.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small ParticlesWiley-Interscience, New York, 1983), Chap. 4.

J. V. Dave, Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere, IBM Order No. 360D-17.4.002 (IBM Scientific Center, Palo Alto, Calif., 1968).

J. C. Maxwell, A Treatise on Electricity and Magnetism (Dover, New York, 1954) Vol. 1, p. 196.

E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Cambridge U. Press, Cambridge, 1931), pp. 129–132.

W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1962), p. 15.

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 (14)

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

a n = x D n ( m x ) ψ n ( x ) + m n ψ n ( x ) - m x ψ n - 1 ( x ) x D n ( m x ) ξ n ( x ) + m n ξ n ( x ) - m x ξ n - 1 ( x ) ,
b n = m x D n ( m x ) ψ n ( x ) + n ψ n ( x ) - x ψ n - 1 ( x ) m x D n ( m x ) ξ n ( x ) + n ξ n ( x ) - x ξ n - 1 ( x ) ,
D n - 1 = n m x - 1 D n + n / m x ,
z n - 1 = 2 n + 1 x z n - z n + 1 .
a n - 1 = u y + A 3 u z + B 1 y + A 4 z u v + A 3 u w + B 1 v + A 4 w ,
b n - 1 = u y + B 1 y + A 2 z u v + B 1 v + A 2 w ,
a n = u y + A 1 y + A 2 z u v + A 1 v + A 2 w ,
b n = u y + B 1 y + B 2 z u v + B 1 v + B 2 w ,
a n + 1 = u y + A 5 u z + A 6 y + A 7 z u v + A 5 u w + A 6 v + A 7 w ,
b n + 1 = u y + B 3 y + A 2 z u v + B 3 v + A 2 w ,
A 1 = m n x ,             A 2 = - m ,             B 1 = n / m x ,             B 2 = - 1 / m , A 3 = n ( 1 - m 2 ) m 2 x ,             A 4 = n 2 - m 2 x 2 - m 2 n 2 m 3 x 2 , A 5 = x ( m 2 - 1 ) ( n + 1 ) ( n + 1 ) ( 2 n + 1 ) ( 1 - m 2 ) + m 2 x 2 , A 6 = n m 2 x + ( n + 1 ) 2 ( 2 n + 1 ) ( m 2 - 1 ) m x [ ( n + 1 ) ( 2 n + 1 ) ( 1 - m 2 ) + m 2 x 2 ] , A 7 = ( n + 1 ) 2 ( 1 - m 2 ) - m 2 x 2 m [ ( n + 1 ) ( 2 n + 1 ) ( 1 - m 2 ) + m 2 x 2 ] , B 3 = ( 2 n + 1 ) m 2 - ( n + 1 ) m x .
[ a n ( A 1 + u ) a n A 2 - ( A 1 + u ) - A 2 b n ( B 1 + u ) b n B 2 - ( B 1 + u ) - B 2 b n - 1 ( B 1 + u ) b n - 1 A 2 - ( B 1 + u ) - A 2 a n - 1 ( B 1 + u ) a n - 1 ( A 3 u + A 4 ) - ( B 1 + u ) - ( A 3 u + A 4 ) ] [ v w y z ] = 0.
u 2 - ( A 1 Δ + A 2 - A 4 A 3 ) u + A 1 ( B 2 - A 4 ) A 3 Δ = 0 ,
Δ = ( a n - b n ) ( a n - 1 - b n - 1 ) ( b n - a n - 1 ) ( a n - b n - 1 ) .

Metrics