Abstract

Efficient, numerically stable, methods for the calculation of light-scattering intensity functions for concentrically coated spheres are discussed. Earlier forms of these equations are subject to various numerical difficulties which give rise to significant errors, especially for thin absorbing shells. The present equations are accurate for all refractive indices, for large and small particles, and for cores with any relative size.

© 1981 Optical Society of America

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References

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  1. A. L. Aden, M. Kerker, J. Appl. Phys. 22, 1242 (1951).
    [CrossRef]
  2. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  3. W. F. Espenscheid, E. Willis, E. Matijevic, M. Kerker, J. Colloid Sci. 20, 501 (1965).
    [CrossRef] [PubMed]
  4. G. W. Kattawar, D. A. Hood, Appl. Opt. 15, 1996 (1976).
    [CrossRef] [PubMed]
  5. T. P. Ackerman, O. B. Toon, Appl. Opt., 20, 3661 (1981).
    [CrossRef] [PubMed]
  6. J. V. Dave, Report 320-3236, IBM Scientific Center, Palo Alto, Calif. (1968).
  7. J. V. Dave, Appl. Opt. 8, 155 (1969).
    [CrossRef] [PubMed]
  8. W. J. Wiscombe, NCAR/TN-140 (1979); Appl. Opt. 19, 1505 (1980).
    [PubMed]
  9. G. W. Kattawar, G. N. Plass, Appl. Opt. 6, 1377 (1967).
    [CrossRef] [PubMed]

1981 (1)

1979 (1)

W. J. Wiscombe, NCAR/TN-140 (1979); Appl. Opt. 19, 1505 (1980).
[PubMed]

1976 (1)

1969 (1)

1967 (1)

1965 (1)

W. F. Espenscheid, E. Willis, E. Matijevic, M. Kerker, J. Colloid Sci. 20, 501 (1965).
[CrossRef] [PubMed]

1951 (1)

A. L. Aden, M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

Ackerman, T. P.

Aden, A. L.

A. L. Aden, M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

Dave, J. V.

J. V. Dave, Appl. Opt. 8, 155 (1969).
[CrossRef] [PubMed]

J. V. Dave, Report 320-3236, IBM Scientific Center, Palo Alto, Calif. (1968).

Espenscheid, W. F.

W. F. Espenscheid, E. Willis, E. Matijevic, M. Kerker, J. Colloid Sci. 20, 501 (1965).
[CrossRef] [PubMed]

Hood, D. A.

Kattawar, G. W.

Kerker, M.

W. F. Espenscheid, E. Willis, E. Matijevic, M. Kerker, J. Colloid Sci. 20, 501 (1965).
[CrossRef] [PubMed]

A. L. Aden, M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

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

Matijevic, E.

W. F. Espenscheid, E. Willis, E. Matijevic, M. Kerker, J. Colloid Sci. 20, 501 (1965).
[CrossRef] [PubMed]

Plass, G. N.

Toon, O. B.

Willis, E.

W. F. Espenscheid, E. Willis, E. Matijevic, M. Kerker, J. Colloid Sci. 20, 501 (1965).
[CrossRef] [PubMed]

Wiscombe, W. J.

W. J. Wiscombe, NCAR/TN-140 (1979); Appl. Opt. 19, 1505 (1980).
[PubMed]

Appl. Opt. (4)

J. Appl. Phys. (1)

A. L. Aden, M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

J. Colloid Sci. (1)

W. F. Espenscheid, E. Willis, E. Matijevic, M. Kerker, J. Colloid Sci. 20, 501 (1965).
[CrossRef] [PubMed]

NCAR/TN-140 (1)

W. J. Wiscombe, NCAR/TN-140 (1979); Appl. Opt. 19, 1505 (1980).
[PubMed]

Other (2)

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

J. V. Dave, Report 320-3236, IBM Scientific Center, Palo Alto, Calif. (1968).

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Equations (27)

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Q ext = ( 2 α 2 ) n = 1 ( 2 n + 1 ) [ Re ( a n + b n ) ] ,
a n = ψ n ( z 2 ) ζ n ( z 2 ) [ U 1 ( k 1 + U 2 U 3 ) - k 3 U 2 U 4 U 5 ( k 1 + U 2 U 3 ) - k 3 U 2 U 4 ] ,
b n = ψ n ( z 2 ) ζ n ( z 2 ) [ U 6 ( k 2 + U 7 U 3 ) - k 3 U 7 U 4 U 8 ( k 2 + U 7 U 3 ) - k 2 U 7 U 4 ] ,
U 1 = k 3 η n 1 ( z 1 ) - k 2 η n 1 ( z 2 ) = { [ k 3 k 2 η n 1 ( z 1 ) + n z 2 ] ψ n ( z 2 ) - ψ n - 1 ( z 2 ) } k 2 ψ n ( z 2 ) ,
U 2 = k 1 η n 1 ( z 4 ) - k 2 η n 1 ( z 3 ) ,
U 3 = - i [ ζ n ( z 1 ) ψ n ( z 1 ) ψ n ( z 4 ) - ζ n ( z 4 ) ] ψ n ( z 4 ) ,
U 4 = [ ψ n ( z 4 ) / ψ n ( z 1 ) ] 2 ,
U 5 = k 3 η n 1 ( z 1 ) - k 2 η n 3 ( z 2 ) = { [ k 3 k 2 η n 1 ( z 1 ) + n z 2 ] ζ n ( z 2 ) - ζ n - 1 ( z 2 ) } k 2 ζ n ( z 2 ) ,
U 6 = k 2 η n 1 ( z 1 ) - k 3 η n 3 ( z 2 ) = { [ k 2 k 3 η n 1 ( z 1 ) + n z 2 ] ψ n ( z 2 ) - ψ n - 1 ( z 2 ) } k 3 ζ n ( z 2 ) ,
U 7 = k 2 η n 1 ( z 4 ) - k 1 η n 1 ( z 3 ) ,
U 8 = k 2 η n 1 ( z 1 ) - k 3 η n 3 ( z 2 ) = { [ k 2 k 3 η n 1 ( z 1 ) + n z 2 ] ζ n ( z 2 ) - ζ n - 1 ( z 2 ) } k 3 ζ ( z 2 ) ,
ψ n ( z ) = z j n ( z ) ;             j n ( z ) = spherical Bessel function , ζ n ( z ) = z h n 2 ( z ) ;             h n 2 = spherical Hankel function , η n 1 ( z ) = ψ n ( z ) / ψ n ( z ) , η n 3 ( z ) = ζ n ( z ) / ζ n ( z ) ,
k 1 = m c 2 π / λ , k 2 = m s 2 π / λ , k 3 = 2 π / λ , z 1 = k 2 R s , z 2 = k 3 R s , z 3 = k 1 R c , z 4 = k 2 R c , m c = refractive index of core , m s = refractive index of shell , R c = radius of core , R s = radius of entire particle .
a n = a n h { 1 + U 2 [ U 3 k 2 - k 3 k 2 ψ n ( z 2 ) U 4 U 1 ] 1 + U 2 [ U 3 k 2 - k 3 k 2 ζ n ( z 2 ) U 4 U 5 ] } ,
b n = b n h { 1 + U 7 [ U 3 k 3 - k 2 k 3 ψ n ( z 2 ) U 4 U 6 ] 1 + U 7 [ U 3 k 3 - k 2 k 3 ζ n ( z 2 ) U 4 U 8 ] } .
η n 1 = n + 1 z - 1 ( n + 1 z + η n + 1 1 ) .
ψ n ( z 2 ) = Re [ ζ n ( z 2 ) ] ζ n ( z 2 ) = 2 n - 1 z 2 ζ n - 1 ( z 2 ) - ζ n - 2 ( z 2 ) . ζ - 1 ( z 2 ) = cos z 2 - i sin z 2 ζ 0 ( z 2 ) = sin ( z 2 ) + i cos z 2
ψ n ( z 4 ) ψ n ( z 1 ) = ψ n - 1 ( z 4 ) ψ n - 1 ( z 1 ) z 4 [ η n 1 ( z 1 ) + n / z 1 z 4 η n 1 ( z 4 ) + n ]
ψ 0 ( z 4 ) ψ 0 ( z 1 ) = A sin x 4 + B i cos x 4 C sin x 1 + D i cos x 1 ,
z 4 = x 4 + i y 4 , z 1 = x 1 + i y 1 , A = exp ( 2 y 4 + y 1 ) + exp ( y 1 ) , B = exp ( 2 y 4 + y 1 ) - exp ( y 1 ) , C = exp ( 2 y 1 + y 4 ) + exp ( y 4 ) . D = exp ( 2 y 1 + y 4 ) - exp ( y 4 ) .
n + ψ n ( z 4 ) ψ n ( z 1 ) = [ ψ n - 1 ( z 4 ) ψ n - 1 ( z 1 ) + n - 1 ] [ η n 1 ( z 1 ) + n / z 1 η n 1 ( z 4 ) + n / z 4 ] ,
n = n - 1 [ η n 1 ( z 1 ) + n / z 1 η n 1 ( z 4 ) + n / z 4 ] .
ζ n ( z 1 ) ψ n ( z 4 ) = ζ n - 1 ( z 1 ) ψ n - 1 ( z 4 ) z 4 [ η n 3 ( z 1 ) + n / z 1 ] [ z 4 η n 1 ( z 4 ) + n ] ,
ζ n ( z 4 ) ψ n ( z 4 ) = ζ n - 1 ( z 4 ) ψ n - 1 ( z 4 ) z 4 2 [ z 4 η n 3 ( z 4 ) + n ] [ z 4 η n 1 ( z 4 ) + n ] ,
ζ 0 ( z 4 ) ψ 0 ( z 4 ) = ½ + ( sin 2 x 4 - ½ + i cos x 4 sin x 4 ) exp ( 2 y 4 ) ζ 0 ( z 1 ) ψ 0 ( z 4 ) = ½ [ ( sin x 1 sin z 4 - cos x 1 cos x 4 ) + i ( sin x 4 cos x 1 + cos x 4 sin x 1 ) ] exp ( y 1 + y 4 ) + ½ [ ( sin x 1 sin x 4 + cos x 1 cos x 4 ) + i ( sin x 4 cos x 1 - cos x 4 sin x 1 ) ] exp ( y 1 - y 4 ) }
η n 3 ( z ) = - n z + [ n z - η n - 1 3 ( z ) ] - 1 ,
η 0 3 ( z ) = - i .

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