Abstract

Scattering functions have been calculated for a homogeneous sphere surrounded by a concentric spherical shell. The scattering is insensitive to radial variation of the refractive index in the shell. The scattering from the radially varying case can be approximated by a sphere with a homogeneous concentric shell whose refractive index is intermediate between the internal and the external media.

© 1966 Optical Society of America

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References

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  1. Y. Nomura and K. Takaku, Tohoku Research Institutes, Res. Inst. Elec. Commun. 7B, 107 (1955).
  2. S. Levine and M. Kerker, in Electromagnetic Scattering, M. Kerker, Ed. (Pergamon Press, Oxford, 1963), pp. 37–46.
  3. A. L. Aden and M. Kerker, J. Appl. Phys. 22, 1242 (1951).
    [Crossref]
  4. M. Kerker, J. P. Kratohvil, and E. Matiiević, J. Opt. Soc. Am. 52, 551 (1962).
    [Crossref]
  5. W. F. Espenscheid, E. Willis, E. Matijević, and M. Kerker, J. Colloid Sci. 20, 501 (1965).
    [Crossref] [PubMed]
  6. A. Güttler, Ann. Physik 11, 65 (1952).
    [Crossref]
  7. H. C. van de Hulst, Light Scattering bv Small Particles (John Wiley & Sons, New York, 1957), p. 74.
  8. M. Kerker, Discussions Faraday Soc. 30, 223 (1960).

1965 (1)

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

1962 (1)

1960 (1)

M. Kerker, Discussions Faraday Soc. 30, 223 (1960).

1955 (1)

Y. Nomura and K. Takaku, Tohoku Research Institutes, Res. Inst. Elec. Commun. 7B, 107 (1955).

1952 (1)

A. Güttler, Ann. Physik 11, 65 (1952).
[Crossref]

1951 (1)

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

Aden, A. L.

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

Espenscheid, W. F.

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

Güttler, A.

A. Güttler, Ann. Physik 11, 65 (1952).
[Crossref]

Kerker, M.

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

M. Kerker, J. P. Kratohvil, and E. Matiiević, J. Opt. Soc. Am. 52, 551 (1962).
[Crossref]

M. Kerker, Discussions Faraday Soc. 30, 223 (1960).

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

S. Levine and M. Kerker, in Electromagnetic Scattering, M. Kerker, Ed. (Pergamon Press, Oxford, 1963), pp. 37–46.

Kratohvil, J. P.

Levine, S.

S. Levine and M. Kerker, in Electromagnetic Scattering, M. Kerker, Ed. (Pergamon Press, Oxford, 1963), pp. 37–46.

Matiievic, E.

Matijevic, E.

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

Nomura, Y.

Y. Nomura and K. Takaku, Tohoku Research Institutes, Res. Inst. Elec. Commun. 7B, 107 (1955).

Takaku, K.

Y. Nomura and K. Takaku, Tohoku Research Institutes, Res. Inst. Elec. Commun. 7B, 107 (1955).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering bv Small Particles (John Wiley & Sons, New York, 1957), p. 74.

Willis, E.

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

Ann. Physik (1)

A. Güttler, Ann. Physik 11, 65 (1952).
[Crossref]

Discussions Faraday Soc. (1)

M. Kerker, Discussions Faraday Soc. 30, 223 (1960).

J. Appl. Phys. (1)

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

J. Colloid Sci. (1)

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

J. Opt. Soc. Am. (1)

Tohoku Research Institutes, Res. Inst. Elec. Commun. (1)

Y. Nomura and K. Takaku, Tohoku Research Institutes, Res. Inst. Elec. Commun. 7B, 107 (1955).

Other (2)

S. Levine and M. Kerker, in Electromagnetic Scattering, M. Kerker, Ed. (Pergamon Press, Oxford, 1963), pp. 37–46.

H. C. van de Hulst, Light Scattering bv Small Particles (John Wiley & Sons, New York, 1957), p. 74.

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Figures (1)

F. 1
F. 1

Refractive-index profiles. The variation of the concentric shell region between χ=α and β is given by n2=Aχm. For values of A and m corresponding to the above four cases, see Table I.

Tables (4)

Tables Icon

Table I Values of parameters A and m for which calculations have been obtained.

Tables Icon

Table II Scattering coefficients for α=0.18, β=0.20.

Tables Icon

Table III Angular intensity functions I1 for α=4.5, β=5.0.

Tables Icon

Table IV Angular intensity functions I2 for α=4.5, β=5.0.

Equations (19)

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n 2 = A χ m .
χ = 2 π r / λ ,
I 1 = λ 2 i 1 4 π 2 r 2 = λ 2 4 π 2 r 2 | l = 1 2 l + 1 l ( l + 1 ) { a l P l 1 ( cos θ ) sin θ + b l d d θ P l 1 ( cos θ ) } | 2
I 2 = λ 2 i 2 4 π 2 r 2 = λ 2 4 π 2 r 2 | l = 1 2 l + 1 l ( l + 1 ) { b l P l 1 ( cos θ ) sin θ + a l d d θ P l 1 ( cos θ ) } | 2 ,
a l = α m J ν ( X α ) α m J ν ( X α ) n 1 1 2 J l + 1 2 ( n 1 α ) 0 β m J ν ( X β ) β m J ν ( X β ) 0 J l + 1 2 ( β ) V l ( α ) / [ n 2 ( α ) ] 2 V ̂ l ( α ) / [ n 2 ( α ) ] 2 f l ( α ) / n 1 2 0 V l ( β ) / [ n 2 ( β ) ] 2 V ̂ l ( β ) / [ n 2 ( β ) ] 2 0 g l ( β ) Other elements of this determinant are the same as in the numerator 0 H l + 1 2 ( 2 ) ( β ) 0 h l ( β )
b l = J μ ( X α ) J μ ( X α ) n 1 1 2 J l + 1 2 ( n 1 α ) 0 J μ ( X β ) J μ ( X β ) 0 J l + 1 2 ( β ) Q l ( α ) Q ̂ l ( α ) f l ( α ) 0 Q l ( β ) Q ̂ l ( β ) 0 g l ( β ) Other elements of this determinant are the same as in the numerator 0 H l + 1 2 ( 2 ) ( β ) 0 h l ( β )
μ = ( 2 l + 1 ) / 2 ( m + 1 )
ν = [ 1 / ( m + 1 ) ] [ l ( l + 1 ) + ( m + 1 / 2 ) 2 ] 1 2
X χ = [ A / ( m + 1 ) ] χ m + 1
V l ( χ ) = d d χ [ χ m + 1 ( π / 2 χ ) 1 2 J ν ( X χ ) ]
V ̂ l ( χ ) = d d χ [ χ m + 1 ( π / 2 χ ) 1 2 J ν ( X χ ) ]
Q l ( χ ) = d d χ [ χ ( π χ / 2 ) 1 2 J μ ( X χ ) ]
Q ̂ l ( χ ) = d d χ [ χ ( π χ / 2 ) 1 2 J μ ( X χ ) ]
f l ( χ ) = d d χ [ χ z l ( 1 ) ( n 1 χ ) ]
g l ( χ ) = d d χ [ χ z l ( 1 ) ( χ ) ]
h l ( χ ) = d d χ [ χ z l ( 3 ) ( χ ) ]
z l ( 1 ) ( χ ) = ( π / 2 χ ) 1 2 J l + 1 2 ( χ )
z l ( 3 ) ( χ ) = ( π / 2 χ ) 1 2 H l + 1 2 ( 2 ) ( χ ) ,
I 1 = ( λ 2 / 9 π 2 r 2 ) | a 1 | 2 ; I 2 = ( λ 2 / 9 π 2 r 2 ) | a 1 | 2 cos 2 θ .