Abstract

The solution for the scattering of radiant energy in the form of electromagnetic waves by concentric isotropic infinitely long circular cylinders is given for the incident energy traveling perpendicular to the cylinder axis.

© 1961 Optical Society of America

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References

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  1. H. C. Van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957), Chap. 15.
  2. B. K. Larkin, Ph.D. thesis (The University of Michigan, Ann Arbor, Michigan, 1957); B. K. Larkin and S. W. Churchill, J. Opt. Soc. Am. 49, 188 (1959). Note footnote 7 in work cited in reference 3 for an error in the latter paper.
    [Crossref]
  3. E. Matijević, R. H. Ottewill, and M. Kerker, J. Opt. Soc. Am. 51, 87 (1961).
    [Crossref]
  4. G. Mie, Ann. Physik 25, 377 (1908).
    [Crossref]
  5. A. L. Aden and M. Kerker, J. Appl. Phys. 22, 1242 (1951).
    [Crossref]
  6. M. Kerker, P. Langleben, and K. L. S. Gunn, J. Meteorol. 8, 424 (1951).
    [Crossref]
  7. D. Atlas, M. Kerker, and W. Hitschfeld, J. Atmospheric Terrest. Phys. 3, 108 (1953).
    [Crossref]
  8. L. J. Battan, Radar Meteorology (The University of Chicago Press, Chicago, Illinois, 1959), Chap. 5.
  9. H. Scharfman, J. Appl. Phys. 25, 1053 (1954).
  10. A. Güttler, Ann. Physik 11, 65 (1952).
    [Crossref]
  11. R. J. Wernick, “On the scattering of electromagnetic waves from concentric spheres,” (Army Chemical Center, Maryland).

1961 (1)

1954 (1)

H. Scharfman, J. Appl. Phys. 25, 1053 (1954).

1953 (1)

D. Atlas, M. Kerker, and W. Hitschfeld, J. Atmospheric Terrest. Phys. 3, 108 (1953).
[Crossref]

1952 (1)

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

1951 (2)

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

M. Kerker, P. Langleben, and K. L. S. Gunn, J. Meteorol. 8, 424 (1951).
[Crossref]

1908 (1)

G. Mie, Ann. Physik 25, 377 (1908).
[Crossref]

Aden, A. L.

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

Atlas, D.

D. Atlas, M. Kerker, and W. Hitschfeld, J. Atmospheric Terrest. Phys. 3, 108 (1953).
[Crossref]

Battan, L. J.

L. J. Battan, Radar Meteorology (The University of Chicago Press, Chicago, Illinois, 1959), Chap. 5.

Gunn, K. L. S.

M. Kerker, P. Langleben, and K. L. S. Gunn, J. Meteorol. 8, 424 (1951).
[Crossref]

Güttler, A.

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

Hitschfeld, W.

D. Atlas, M. Kerker, and W. Hitschfeld, J. Atmospheric Terrest. Phys. 3, 108 (1953).
[Crossref]

Kerker, M.

E. Matijević, R. H. Ottewill, and M. Kerker, J. Opt. Soc. Am. 51, 87 (1961).
[Crossref]

D. Atlas, M. Kerker, and W. Hitschfeld, J. Atmospheric Terrest. Phys. 3, 108 (1953).
[Crossref]

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

M. Kerker, P. Langleben, and K. L. S. Gunn, J. Meteorol. 8, 424 (1951).
[Crossref]

Langleben, P.

M. Kerker, P. Langleben, and K. L. S. Gunn, J. Meteorol. 8, 424 (1951).
[Crossref]

Larkin, B. K.

B. K. Larkin, Ph.D. thesis (The University of Michigan, Ann Arbor, Michigan, 1957); B. K. Larkin and S. W. Churchill, J. Opt. Soc. Am. 49, 188 (1959). Note footnote 7 in work cited in reference 3 for an error in the latter paper.
[Crossref]

Matijevic, E.

Mie, G.

G. Mie, Ann. Physik 25, 377 (1908).
[Crossref]

Ottewill, R. H.

Scharfman, H.

H. Scharfman, J. Appl. Phys. 25, 1053 (1954).

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957), Chap. 15.

Wernick, R. J.

R. J. Wernick, “On the scattering of electromagnetic waves from concentric spheres,” (Army Chemical Center, Maryland).

Ann. Physik (2)

G. Mie, Ann. Physik 25, 377 (1908).
[Crossref]

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

J. Appl. Phys. (2)

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

H. Scharfman, J. Appl. Phys. 25, 1053 (1954).

J. Atmospheric Terrest. Phys. (1)

D. Atlas, M. Kerker, and W. Hitschfeld, J. Atmospheric Terrest. Phys. 3, 108 (1953).
[Crossref]

J. Meteorol. (1)

M. Kerker, P. Langleben, and K. L. S. Gunn, J. Meteorol. 8, 424 (1951).
[Crossref]

J. Opt. Soc. Am. (1)

Other (4)

L. J. Battan, Radar Meteorology (The University of Chicago Press, Chicago, Illinois, 1959), Chap. 5.

H. C. Van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957), Chap. 15.

B. K. Larkin, Ph.D. thesis (The University of Michigan, Ann Arbor, Michigan, 1957); B. K. Larkin and S. W. Churchill, J. Opt. Soc. Am. 49, 188 (1959). Note footnote 7 in work cited in reference 3 for an error in the latter paper.
[Crossref]

R. J. Wernick, “On the scattering of electromagnetic waves from concentric spheres,” (Army Chemical Center, Maryland).

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Figures (1)

Fig. 1
Fig. 1

Coordinates and vectors for scattering by concentric cylinder; a and b are radii of the inner and outer cylinders. Here m0, m1, m2 are the refractive indexes in the indicated regions and S, E, and H are Poynting’s, the electric, and the magnetic vectors, respectively.

Equations (30)

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( r > b ) u = n = F n { J n ( m 0 k r ) b n 0 H n ( m 0 k r ) } ,
( b > r > a ) u = n = F n { B n 1 J n ( m 1 k r ) b n 1 H n ( m 1 k r ) } ,
( r < a ) u = n = F n { B n 2 J n ( m 2 k r ) } .
( r > b ) υ = n = F n { J n ( m 0 k r ) a n 0 H n ( m 0 k r ) } ,
( b > r > a ) υ = n = F n { A n 1 J n ( m 1 k r ) a n 1 H n ( m 1 k r ) } ,
( r < a ) υ = n = F n { A n 2 J n ( m 2 k r ) } .
H n ( Z ) = J n ( Z ) i N n ( Z ) ,
F n = ( 1 ) n e i n θ + i ω t ,
b n 0 m 0 H n ( m 0 α 1 ) b n 1 m 1 H n ( m 1 α 1 ) + B n 1 m 1 J n ( m 1 α 1 ) + 0 = m 0 J n ( m 0 α 1 )
b n 0 m 0 2 H n ( m 0 α 1 ) b n 1 m 1 2 H n ( m 1 α 1 ) + B n 1 m 1 2 J n ( m 1 α 1 ) + 0 = m 0 2 J n ( m 0 α 1 )
0 b n 1 m 1 H n ( m 1 α 2 ) + B n 1 m 1 J n ( m 1 α 2 ) B n 2 m 2 J n ( m 2 α 2 ) = 0
0 b n 1 m 1 2 H n ( m 1 α 2 ) + B n 1 m 1 2 J n ( m 1 α 2 ) B n 2 m 2 2 J n ( m 2 α 2 ) = 0
a n 0 m 0 2 H n ( m 0 α 1 ) a n 1 m 1 2 H n ( m 1 α 1 ) + A n 1 m 1 2 J n ( m 1 α 1 ) + 0 = m 0 2 J n ( m 0 α 1 )
a n 0 m 0 H n ( m 0 α 1 ) a n 1 m 1 H n ( m 1 α 1 ) + A n 1 m 1 J n ( m 1 α 1 ) + 0 = m 0 J n ( m 0 α 1 )
0 a n 1 m 1 2 H n ( m 1 α 2 ) + A n 1 m 1 2 J n ( m 1 α 2 ) A n 2 m 2 2 J n ( m 2 α 2 ) = 0
0 a n 1 m 1 H n ( m 1 α 2 ) + A n 1 m 1 J n ( m 1 α 2 ) A n 2 m 2 J n ( m 2 α 2 ) = 0.
b n 0 = | J n ( m 0 α 1 ) H n ( m 1 α 1 ) J n ( m 1 α 1 ) 0 m 0 J n ( m 0 α 1 ) m 1 H n ( m 1 α 1 ) m 1 J n ( m 1 α 1 ) 0 0 H n ( m 1 α 2 ) J n ( m 1 α 2 ) J n ( m 2 α 2 ) 0 m 1 H n ( m 1 α 2 ) m 1 J n ( m 1 α 2 ) m 2 J n ( m 2 α 2 ) | | H n ( m 0 α 1 ) H n ( m 1 α 1 ) J n ( m 1 α 1 ) 0 m 0 H n ( m 0 α 1 ) m 1 H n ( m 1 α 1 ) m 1 J n ( m 1 α 1 ) 0 0 H n ( m 1 α 2 ) J n ( m 1 α 2 ) J n ( m 2 α 2 ) 0 m 1 H n ( m 1 α 2 ) m 1 J n ( m 1 α 2 ) m 2 J n ( m 2 α 2 ) |
a n 0 = | m 0 J n ( m 0 α 1 ) m 1 H n ( m 1 α 1 ) m 1 J n ( m 1 α 1 ) 0 J n ( m 0 α 1 ) H n ( m 1 α 1 ) J n ( m 1 α 1 ) 0 0 m 1 H n ( m 1 α 2 ) m 1 J n ( m 1 α 2 ) m 2 J n ( m 2 α 2 ) 0 H n ( m 1 α 2 ) J n ( m 1 α 2 ) J n ( m 2 α 2 ) | | m 0 H n ( m 0 α 1 ) m 1 H n ( m 1 α 1 ) m 1 J n ( m 1 α 1 ) 0 H n ( m 0 α 1 ) H n ( m 1 α 1 ) J n ( m 1 α 1 ) 0 0 m 1 H n ( m 1 α 2 ) m 1 J n ( m 1 α 2 ) m 2 J n ( m 2 α 2 ) 0 H n ( m 1 α 2 ) J n ( m 1 α 2 ) J n ( m 2 α 2 ) | .
b n 0 = | J n ( m 0 α ) J n ( m 1 α ) m 0 J n ( m 0 α ) m 1 J n ( m 1 α ) | | H n ( m 0 α ) J n ( m 1 α ) m 0 H n ( m 0 α ) m 1 J n ( m 1 α ) |
a n 0 = | m 0 J n ( m 0 α ) m 1 J n ( m 1 α ) J n ( m 0 α ) J n ( m 1 α ) | | m 0 H n ( m 0 α ) m 1 J n ( m 1 α ) H n ( m 0 α ) J n ( m 1 α ) | .
T 1 ( θ ) = n = b n 0 e i n θ = b 0 0 + 2 n = 1 b n 0 cos n θ
T 2 ( θ ) = n = B n a n 0 e i n θ = a 0 0 + 2 n = 1 a n 0 cos n θ .
b n 0 = b n 0 , a n 0 = a n 0 .
b n j 1 m j 1 H n ( m j 1 α j ) δ ¯ 1 B n j 1 m j 1 J n ( m j 1 α j ) δ ¯ j b n j m j H n ( m j α j ) + B n j m j J n ( m j α j ) = δ 1 m j 1 J n ( m j 1 α j )
b n j 1 m j 1 2 H n ( m j 1 α j ) δ ¯ 1 B n j 1 m j 1 2 J n ( m j 1 α j ) δ ¯ j b n j m j 2 H n ( m j α j ) + B n j m j 2 J n ( m j α j ) = δ 1 m j 1 2 J n ( m j 1 α j )
a n j 1 m j 1 2 H n ( m j 1 α j ) δ ¯ 1 A n j 1 m j 1 2 J n ( m j 1 α j ) δ ¯ j a n j m j 2 H n ( m j α j ) + A n j m j 2 J n ( m j α j ) = δ 1 m j 1 2 J n ( m j 1 α j )
a n j 1 m j 1 H n ( m j 1 α j ) δ ¯ 1 A n j 1 m j 1 J n ( m j 1 α j ) δ ¯ j a n j m j H n ( m j α j ) + A n j m j J n ( m j α j ) = δ 1 m j 1 J n ( m j 1 α j ) ,
δ 1 = 1 when   j = 1 ; δ 1 = 0 when   j 1
δ ¯ 1 = 0 when   j = 1 ; δ ¯ 1 = 1 when   j 1
δ ¯ j = 0 when   j = k ; δ ¯ j = 1 when   j k .