Abstract

The differential scattering characteristics of closed three-dimensional dielectric objects are theoretically investigated. The scattering problem is solved in a spherical basis by the Extended Boundary Condition Method (EBCM) which results in a system of linear equations for the expansion coefficients of the scattered field in terms of the incident field coefficients. The equations are solved numerically for dielectric spheres, spheroids, and finite cylinders to study the dependence of the differential scattering on the size, shape, and index of refraction of the scattering object. The method developed here appears to be most applicable to objects whose physical size is on the order of the wavelength of the incident radiation.

© 1975 Optical Society of America

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References

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  1. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  2. R. G. Kouyoumjian, L. Peters, D. T. Thomas, IEEE Trans. Antennas Propag. AP-11, 690 (1963).
    [CrossRef]
  3. J. B. Keller, J. Opt. Soc. Am. 52, 116 (1962).
    [CrossRef]
  4. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).
  5. C. Yeh, J. Opt. Soc. Am. 55, 309 (1965).
    [CrossRef]
  6. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  7. R. H. T. Bates, J. R. James, I. N. L. Gallett, R. F. Millar, Radio Electron. Eng. 43, 193 (1973).
    [CrossRef]
  8. D. R. Wilton, R. Mittra, IEEE Trans. Antennas Propag. AP-20, 310 (1972).
    [CrossRef]
  9. C. Yeh, Phys. Rev. 135, A1193 (31August1964).
    [CrossRef]
  10. J. H. Richmond, IEEE Trans. Antennas Propag. AP-13, 334 (1965).
    [CrossRef]
  11. J. H. Richmond, Proc. IEEE 53, 796 (1965).
    [CrossRef]
  12. P. C. Waterman, Proc. IEEE 53, 805 (1965).
    [CrossRef]
  13. P. C. Waterman, C. V. McCarthy, “Numerical Solution of Electromagnetic Scattering Problems,” Mitre Corporation, Bedford, Massachusetts, Report MTP-74 (N69-31912) (June1968).
  14. P. C. Waterman, Alta Freq. 38 (Speciale), 348 (1969).
  15. P. C. Waterman, Phys. Rev. D 3, 825 (1971).
    [CrossRef]
  16. H. Honl, A. W. Maue, K. Westpfahl, Handbuch der Physik (Springer Verlag, Berlin, 1961), Vol. 25/1.
  17. S. A. Schelkunoff, Electromagnetic Waves (D. Von Nostrand, New York, 1943).
  18. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  19. A. Nehari, Introduction to Complex Analysis (Allyn and Bacon, Boston, Mass., 1968).

1973

R. H. T. Bates, J. R. James, I. N. L. Gallett, R. F. Millar, Radio Electron. Eng. 43, 193 (1973).
[CrossRef]

1972

D. R. Wilton, R. Mittra, IEEE Trans. Antennas Propag. AP-20, 310 (1972).
[CrossRef]

1971

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

1969

P. C. Waterman, Alta Freq. 38 (Speciale), 348 (1969).

1965

C. Yeh, J. Opt. Soc. Am. 55, 309 (1965).
[CrossRef]

J. H. Richmond, IEEE Trans. Antennas Propag. AP-13, 334 (1965).
[CrossRef]

J. H. Richmond, Proc. IEEE 53, 796 (1965).
[CrossRef]

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

1964

C. Yeh, Phys. Rev. 135, A1193 (31August1964).
[CrossRef]

1963

R. G. Kouyoumjian, L. Peters, D. T. Thomas, IEEE Trans. Antennas Propag. AP-11, 690 (1963).
[CrossRef]

1962

Bates, R. H. T.

R. H. T. Bates, J. R. James, I. N. L. Gallett, R. F. Millar, Radio Electron. Eng. 43, 193 (1973).
[CrossRef]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Gallett, I. N. L.

R. H. T. Bates, J. R. James, I. N. L. Gallett, R. F. Millar, Radio Electron. Eng. 43, 193 (1973).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).

Honl, H.

H. Honl, A. W. Maue, K. Westpfahl, Handbuch der Physik (Springer Verlag, Berlin, 1961), Vol. 25/1.

James, J. R.

R. H. T. Bates, J. R. James, I. N. L. Gallett, R. F. Millar, Radio Electron. Eng. 43, 193 (1973).
[CrossRef]

Keller, J. B.

Kerker, M.

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

Kouyoumjian, R. G.

R. G. Kouyoumjian, L. Peters, D. T. Thomas, IEEE Trans. Antennas Propag. AP-11, 690 (1963).
[CrossRef]

Maue, A. W.

H. Honl, A. W. Maue, K. Westpfahl, Handbuch der Physik (Springer Verlag, Berlin, 1961), Vol. 25/1.

McCarthy, C. V.

P. C. Waterman, C. V. McCarthy, “Numerical Solution of Electromagnetic Scattering Problems,” Mitre Corporation, Bedford, Massachusetts, Report MTP-74 (N69-31912) (June1968).

Millar, R. F.

R. H. T. Bates, J. R. James, I. N. L. Gallett, R. F. Millar, Radio Electron. Eng. 43, 193 (1973).
[CrossRef]

Mittra, R.

D. R. Wilton, R. Mittra, IEEE Trans. Antennas Propag. AP-20, 310 (1972).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Nehari, A.

A. Nehari, Introduction to Complex Analysis (Allyn and Bacon, Boston, Mass., 1968).

Peters, L.

R. G. Kouyoumjian, L. Peters, D. T. Thomas, IEEE Trans. Antennas Propag. AP-11, 690 (1963).
[CrossRef]

Richmond, J. H.

J. H. Richmond, IEEE Trans. Antennas Propag. AP-13, 334 (1965).
[CrossRef]

J. H. Richmond, Proc. IEEE 53, 796 (1965).
[CrossRef]

Schelkunoff, S. A.

S. A. Schelkunoff, Electromagnetic Waves (D. Von Nostrand, New York, 1943).

Stratton, J. A.

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

Thomas, D. T.

R. G. Kouyoumjian, L. Peters, D. T. Thomas, IEEE Trans. Antennas Propag. AP-11, 690 (1963).
[CrossRef]

Waterman, P. C.

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

P. C. Waterman, Alta Freq. 38 (Speciale), 348 (1969).

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

P. C. Waterman, C. V. McCarthy, “Numerical Solution of Electromagnetic Scattering Problems,” Mitre Corporation, Bedford, Massachusetts, Report MTP-74 (N69-31912) (June1968).

Westpfahl, K.

H. Honl, A. W. Maue, K. Westpfahl, Handbuch der Physik (Springer Verlag, Berlin, 1961), Vol. 25/1.

Wilton, D. R.

D. R. Wilton, R. Mittra, IEEE Trans. Antennas Propag. AP-20, 310 (1972).
[CrossRef]

Yeh, C.

C. Yeh, J. Opt. Soc. Am. 55, 309 (1965).
[CrossRef]

C. Yeh, Phys. Rev. 135, A1193 (31August1964).
[CrossRef]

Alta Freq.

P. C. Waterman, Alta Freq. 38 (Speciale), 348 (1969).

IEEE Trans. Antennas Propag.

R. G. Kouyoumjian, L. Peters, D. T. Thomas, IEEE Trans. Antennas Propag. AP-11, 690 (1963).
[CrossRef]

D. R. Wilton, R. Mittra, IEEE Trans. Antennas Propag. AP-20, 310 (1972).
[CrossRef]

J. H. Richmond, IEEE Trans. Antennas Propag. AP-13, 334 (1965).
[CrossRef]

J. Opt. Soc. Am.

Phys. Rev.

C. Yeh, Phys. Rev. 135, A1193 (31August1964).
[CrossRef]

Phys. Rev. D

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

Proc. IEEE

J. H. Richmond, Proc. IEEE 53, 796 (1965).
[CrossRef]

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

Radio Electron. Eng.

R. H. T. Bates, J. R. James, I. N. L. Gallett, R. F. Millar, Radio Electron. Eng. 43, 193 (1973).
[CrossRef]

Other

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

P. C. Waterman, C. V. McCarthy, “Numerical Solution of Electromagnetic Scattering Problems,” Mitre Corporation, Bedford, Massachusetts, Report MTP-74 (N69-31912) (June1968).

H. Honl, A. W. Maue, K. Westpfahl, Handbuch der Physik (Springer Verlag, Berlin, 1961), Vol. 25/1.

S. A. Schelkunoff, Electromagnetic Waves (D. Von Nostrand, New York, 1943).

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

A. Nehari, Introduction to Complex Analysis (Allyn and Bacon, Boston, Mass., 1968).

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

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).

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Figures (14)

Fig. 1
Fig. 1

Scattering geometry.

Fig. 2
Fig. 2

The scattering problem. J ¯ i and are M ¯ i the sources of the incident field.

Fig. 3
Fig. 3

Application of the equivalence theorem to the scattered field sources.

Fig. 4
Fig. 4

Application of the equivalence theorem to the negative of the incident field sources.

Fig. 5
Fig. 5

Summing of sources and fields to obtain the external problem: (a) the sum of Figs. 3(b) and 4(b); (b) adding in the incident sources and fields.

Fig. 6
Fig. 6

Region of convergence of the Green's function expansion (shaded).

Fig. 7
Fig. 7

Rayleigh scattering.

Fig. 8
Fig. 8

Sphere scattering, ka = 1.0, r = 4.0.

Fig. 9
Fig. 9

Sphere scattering, vertical polarization, ka = 3.0, 5.0, r = 4.0.

Fig. 10
Fig. 10

3:1 prolate spheroid scattering, vertical polarization, ka = 7.114, r = 5.0: (a) azimuthal plane; (b) equatorial plane.

Fig. 11
Fig. 11

3:1 oblate spheroid scattering, vertical polarization, ka = 4.932, r = 5.0: (a) azimuthal plane; (b) equatorial plane.

Fig. 12
Fig. 12

Cylinder scattering, equatorial plane, r = 1.96, r = 2b: (a) 2:1, ka = 5.288, b/a = 0.577; (b) 3:1, ka = 8.341, b/a = 0.366.

Fig. 13
Fig. 13

Scattering by similar bodies, vertical polarization, azimuthal plane, r = 2.28: (a) 2:1 prolate spheroid, ka = 9.529, a/b = 2.0; (b) 2:1 cylinder, ka = 4.765, a/b = 1.0.

Fig. 14
Fig. 14

Cylinder scattering, azimuthal plane, ka = 4.765, a/b = 1.0, m = 1.51, m = 1.51−j0.05.

Equations (36)

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Ē s = Ē Ē i , H ¯ s = H ¯ H ¯ i .
Ē s = × F ¯ 1 j ω 0 ( × × Ā ) ,
H ¯ s = × Ā 1 j ω μ 0 ( × × F ¯ ) ,
Ā = 1 4 π s J ¯ + exp ( j k | r ¯ r ¯ | ) | r ¯ r ¯ | d S , J ¯ + = n ¯ × H ¯ +
F ¯ = 1 4 π s M ¯ + exp ( j k | r ¯ r ¯ | ) | r ¯ r ¯ | d S , M ¯ + = Ē + × n ¯ ;
Ē s ( r ¯ ) = × s ( n ¯ × Ē + ) g ( k R ) d S × × s 1 j ω 0 ( n ¯ × H ¯ + ) g ( k R ) d S ,
R = | r r | and k = 2 π / λ .
Ē ( r ) 0 } = Ē i ( r ¯ ) + × s ( n ¯ × Ē + ) g ( k R ) d S × × s 1 j ω 0 ( n ¯ × H ¯ + ) g ( k R ) d S ; r ¯ { outside S inside S .
× s ( n ¯ × Ē + ) g ( k R ) d S × × s 1 j ω 0 ( n ¯ × H ¯ + ) g ( k R ) d S = Ē i ( r ¯ ) .
M ¯ σ m n ( r ¯ ) = × r ¯ cos m ϕ sin m ϕ P n m ( cos θ ) z n ( k r )
N ¯ σ m n ( r ¯ ) = 1 k × M ¯ σ m n ( r ¯ ) ,
Ē i ( r ¯ ) = ν = 1 D ν [ a ν M ¯ ν 1 ( k r ¯ ) + b ν N ¯ ν 1 ( k r ¯ ) ] ,
D ν = m ( 2 n + 1 ) ( n m ) ! 4 n ( n + 1 ) ( n + m ) ! , m = 1 2 } , m = 0 > 0 .
( n ¯ × Ē + ) g ( k R ) = ( n ¯ × Ē + ) · G ¯ ¯ ,
( n ¯ × H ¯ + ) g ( k R ) = ( n ¯ × H ¯ + ) · G ¯ ¯ ,
G ¯ ¯ ( k r ) = j k π ν = 1 D ν [ M ¯ ν 3 ( k r ¯ > ) M ¯ ν 1 ( k r ¯ < ) + N ¯ ν 3 ( k r ¯ > ) N ¯ ν 1 ( k r ¯ < ) ] ,
j k 2 π S [ N ¯ ν 3 ( k r ¯ ) · ( n ¯ × E + ¯ ) + j ( μ 0 0 ) 1 / 2 M ¯ ν 3 ( k r ¯ ) · ( n ¯ × H + ¯ ) ] d S = a ν ;
i k 2 π s [ M ¯ ν 3 ( k r ¯ ) · ( n ¯ × Ē + ) + j ( μ 0 0 ) 1 / 2 N ¯ ν 3 ( k r ¯ ) · ( n ¯ × H ¯ + ) ] d s = b ν ;
Ē ( k r ¯ ) = μ = 1 N [ c μ M ¯ μ 1 ( k r ¯ ) + d μ N ¯ μ 1 ( k r ¯ ) ] ,
H ¯ ( k r ¯ ) = 1 j ω μ [ × Ē ( k r ) ] = j ( r μ r ) 1 / 2 ( 0 μ 0 ) 1 / 2 μ = 1 N [ c μ N ¯ μ 1 ( k r ¯ ) + d μ M ¯ μ 1 ( k r ¯ ) ] .
n ¯ × H ¯ + = n ¯ × H ¯
n ¯ × Ē + = n ¯ × Ē .
n ¯ × Ē = μ = 1 N [ c μ n ¯ × M ¯ μ 1 ( k r ¯ ) + d μ n ¯ × N ¯ μ 1 ( k r ¯ ) ] ,
n ¯ × H ¯ = j ( r μ r ) 1 / 2 ( 0 μ 0 ) 1 / 2 μ = 1 N [ c μ n ¯ × N ¯ μ 1 ( k r ¯ ) + d μ n ¯ × M ¯ μ 1 ( k r ¯ ) ] .
[ K + ( r μ r ) 1 / 2 J ] c μ + [ L + ( r μ r ) 1 / 2 I ] d μ = j a ν ν = 1 , 2 , . . . , N ,
[ I + ( r μ r ) 1 / 2 L ] c μ + [ J + ( r μ r ) 1 / 2 K ] d μ = j b ν ,
I = k 2 π S n ¯ · M ¯ ν 3 ( k r ¯ ) × M ¯ μ 1 ( k r ¯ ) d S , J = k 2 π S n ¯ · M ¯ ν 3 ( k r ) × N ¯ μ 1 ( k r ¯ ) d S , K = k 2 π S n ¯ · N ¯ ν 3 ( k r ¯ ) × M ¯ μ 1 ( k r ¯ ) d S , L = k 2 π S n ¯ · N ¯ ν 3 ( k r ¯ ) × N ¯ μ 1 ( k r ¯ ) d S .
Ē s ( k r ¯ ) = ν = 1 N [ p ν M ¯ ν 3 ( k r ¯ ) + q ν N ¯ ν 3 ( k r ¯ ) ] ,
p ν = j D ν μ = 1 N { [ K + ( r μ r ) 1 / 2 J ] c μ + [ L + ( r μ r ) 1 / 2 I ] d μ } ,
q ν = j D ν μ = 1 N { [ I + ( r μ r ) 1 / 2 L ] c μ + [ J + ( r μ r ) 1 / 2 K ] d μ } ,
I = k 2 π S n ¯ · M ¯ ν 1 ( k r ¯ ) × M ¯ μ 1 ( k r ¯ ) d S , J = k 2 π S n ¯ · M ¯ ν 1 ( k r ¯ ) × N ¯ μ 1 ( k r ¯ ) d S , K = k 2 π S n ¯ · N ¯ ν 1 ( k r ¯ ) × M ¯ μ 1 ( k r ¯ ) d S , L = k 2 π S n ¯ · N ¯ ν 1 ( k r ¯ ) × N ¯ μ 1 ( k r ¯ ) d S .
Ē s ( k r ¯ ) = F ¯ ( θ s , ϕ s / θ i , ϕ i ) exp ( j k r ) r , k r ,
σ D = lim r [ 4 π r 2 S s ( θ s , ϕ s ) S i ( θ i , ϕ i ) ] ,
= | F ¯ ( θ s , ϕ s / θ i , ϕ i ) | 2 2 Z 0 r 2 , Z 0 = μ 0 / 0 ,
= | Ē i | 2 2 Z 0 .
σ D ( θ s , ϕ s / θ i , ϕ i ) = 4 π | F ¯ ( θ s , ϕ s / θ i , ϕ i ) | 2 .

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