Abstract

The extended boundary condition method is applied to 3-D electromagnetic scattering problems involving multilayered dielectric objects. The theoretical formulation is reviewed, and the numerical problems are discussed. Angular scattering calculations have been made for bacteria-like layered prolate spheroids with a nucleus-like inner layer and a membrane-like outer layer. Comparing the results with those obtained for corresponding homogeneous models, it is found that the internal structure results in a significant difference in the angular scattering characteristics. Calculations are also made to study the variation of the scattering, absorption, extinction, and backscattering cross sections of an oblate spheroidal model of a melting hailstone as either its shape or size changes. These results indicate that the backscattering cross section is most sensitive to changes in structure.

© 1979 Optical Society of America

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  1. A. L. Aden, M. Kerker, J. Appl. Phys. 22, 1242 (1951).
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  2. A. Güttler, Ann. Phys. 6, 76 (1952).
  3. P. J. Wyatt, Appl. Opt. 7, 1879 (1968).
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  4. D. A. Cross, P. Latimer, Appl. Opt. 11, 1225 (1972).
    [CrossRef] [PubMed]
  5. A. L. Koch, Biochem. Biophys. Acta 51, 429 (1961).
    [CrossRef]
  6. A. L. Koch, J. Theor. Biol. 18, 133 (1968).
    [CrossRef] [PubMed]
  7. P. C. Waterman, Proc. IEEE 53, 796 (1965).
    [CrossRef]
  8. P. C. Waterman, Alta. Freq. 38 (Speciale), 348 (1969).
  9. P. C. Waterman, Phys. Rev. D 3, 825 (1971).
    [CrossRef]
  10. B. Peterson, S. Ström, Phys. Rev. D 10, 2670 (1974).
    [CrossRef]
  11. V. N. Bringi, T. A. Seliga, IEEE Trans. Antenna Propag. AP-25, 575 (1977).
    [CrossRef]
  12. D-S. Wang, Ph.D. dissertation, U. Utah (1978).
  13. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  14. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  15. P. Barber, C. Yeh, Appl. Opt. 14, 2864 (1975).
    [CrossRef] [PubMed]
  16. L. J. Battan, S. R. Browning, B. M. Herman, Tech. Rept. No. 21, Institute of Atmospheric Physics, U. Arizona, Tucson (June1970).
  17. B. M. Herman, L. J. Battan, J. Meteorol. 18, 468 (1961)
    [CrossRef]

1977 (1)

V. N. Bringi, T. A. Seliga, IEEE Trans. Antenna Propag. AP-25, 575 (1977).
[CrossRef]

1975 (1)

1974 (1)

B. Peterson, S. Ström, Phys. Rev. D 10, 2670 (1974).
[CrossRef]

1972 (1)

1971 (1)

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

1969 (1)

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

1968 (2)

1965 (1)

P. C. Waterman, Proc. IEEE 53, 796 (1965).
[CrossRef]

1961 (2)

A. L. Koch, Biochem. Biophys. Acta 51, 429 (1961).
[CrossRef]

B. M. Herman, L. J. Battan, J. Meteorol. 18, 468 (1961)
[CrossRef]

1952 (1)

A. Güttler, Ann. Phys. 6, 76 (1952).

1951 (1)

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

Aden, A. L.

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

Barber, P.

Battan, L. J.

B. M. Herman, L. J. Battan, J. Meteorol. 18, 468 (1961)
[CrossRef]

L. J. Battan, S. R. Browning, B. M. Herman, Tech. Rept. No. 21, Institute of Atmospheric Physics, U. Arizona, Tucson (June1970).

Bringi, V. N.

V. N. Bringi, T. A. Seliga, IEEE Trans. Antenna Propag. AP-25, 575 (1977).
[CrossRef]

Browning, S. R.

L. J. Battan, S. R. Browning, B. M. Herman, Tech. Rept. No. 21, Institute of Atmospheric Physics, U. Arizona, Tucson (June1970).

Cross, D. A.

Feshbach, H.

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

Güttler, A.

A. Güttler, Ann. Phys. 6, 76 (1952).

Herman, B. M.

B. M. Herman, L. J. Battan, J. Meteorol. 18, 468 (1961)
[CrossRef]

L. J. Battan, S. R. Browning, B. M. Herman, Tech. Rept. No. 21, Institute of Atmospheric Physics, U. Arizona, Tucson (June1970).

Kerker, M.

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

Koch, A. L.

A. L. Koch, J. Theor. Biol. 18, 133 (1968).
[CrossRef] [PubMed]

A. L. Koch, Biochem. Biophys. Acta 51, 429 (1961).
[CrossRef]

Latimer, P.

Morse, P. M.

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

Peterson, B.

B. Peterson, S. Ström, Phys. Rev. D 10, 2670 (1974).
[CrossRef]

Seliga, T. A.

V. N. Bringi, T. A. Seliga, IEEE Trans. Antenna Propag. AP-25, 575 (1977).
[CrossRef]

Stratton, J. A.

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

Ström, S.

B. Peterson, S. Ström, Phys. Rev. D 10, 2670 (1974).
[CrossRef]

Wang, D-S.

D-S. Wang, Ph.D. dissertation, U. Utah (1978).

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, 796 (1965).
[CrossRef]

Wyatt, P. J.

Yeh, C.

Alta. Freq. (1)

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

Ann. Phys. (1)

A. Güttler, Ann. Phys. 6, 76 (1952).

Appl. Opt. (3)

Biochem. Biophys. Acta (1)

A. L. Koch, Biochem. Biophys. Acta 51, 429 (1961).
[CrossRef]

IEEE Trans. Antenna Propag. (1)

V. N. Bringi, T. A. Seliga, IEEE Trans. Antenna Propag. AP-25, 575 (1977).
[CrossRef]

J. Appl. Phys. (1)

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

J. Meteorol. (1)

B. M. Herman, L. J. Battan, J. Meteorol. 18, 468 (1961)
[CrossRef]

J. Theor. Biol. (1)

A. L. Koch, J. Theor. Biol. 18, 133 (1968).
[CrossRef] [PubMed]

Phys. Rev. D (2)

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

B. Peterson, S. Ström, Phys. Rev. D 10, 2670 (1974).
[CrossRef]

Proc. IEEE (1)

P. C. Waterman, Proc. IEEE 53, 796 (1965).
[CrossRef]

Other (4)

D-S. Wang, Ph.D. dissertation, U. Utah (1978).

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

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

L. J. Battan, S. R. Browning, B. M. Herman, Tech. Rept. No. 21, Institute of Atmospheric Physics, U. Arizona, Tucson (June1970).

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Figures (6)

Fig. 1
Fig. 1

(a) Structure of a two-layered object. (b) Scattering geometry of a two-layered object oriented at the angles (θp, ϕp) relative to the incident wave. The angle θs defines the scattering direction in the xz plane (ϕs = 0°).

Fig. 2
Fig. 2

Application of the equivalence principle. The original scattering problem is replaced by three subproblems: (a) Ē3, H ¯ 3, outside S2; (b) Ē2, H ¯ 2, between S2 and S1; (c) Ē1, H ¯ 1, inside S1. J ¯ and M ¯ are electric and magnetic surface currents, respectively.

Fig. 3
Fig. 3

Differential scattering cross section (normalized to π a 2 2) for a prolate spheroid with (—) and without (- - -) an inclusion. The axial ratio of the outer prolate spheroid is 3.0. Vertical polarization. (a) An embedded sphere; (b) an embedded prolate spheroid with an axial ratio of 3.125. The particle drawings are to scale.

Fig. 4
Fig. 4

Differential scattering cross section for a prolate spheroid. Vertical polarization. (a) With (—) and without (- - -) a thin surface layer. Normalized to π a 2 2. (b) The layered prolate spheroid in (a) with (—) and without (- - -) a spherical inclusion. Normalized to π a 3 2.

Fig. 5
Fig. 5

(a) The variations of the scattering (sc), extinction (ext), and absorption (abs) efficiency as the shape of the hailstone changes from spherical to oblate spheroidal. (b) The backscattering efficiency as the angle of incidence changes from end-on (1) to broadside with horizontal polarization (2H), and vertical polarization (2V).

Fig. 6
Fig. 6

Scattering and absorption of a hailstone as a function of size. a/b = 1.2, mwater = 8.99–j1.474, mice = 1.77–j0.0024, incident wavelength = 10 cm, thickness of water shell = 1 mm: (a) scattering cross section; (b) absorption cross section; (c) extinction cross section; (d) backscattering cross section. The solid line is for 1:1.2 oblate spheroids and the dotted line, for equivolume spheres.

Equations (23)

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E ¯ 3 ( r ¯ ) 0 } = E ¯ i ( r ¯ ) + × S 2 ( n ¯ 2 × E ¯ + 3 · G ¯ ¯ ( k 3 R ) d s - × × S 2 1 k ω 3 ( n ¯ 2 × H ¯ + 3 ) · G ¯ ¯ ( k 3 R ) d s ; r ¯ { outside S 2 inside S 2 ,
E ¯ 2 ( r ¯ ) 0 } = × S 2 ( - n ¯ 2 × E ¯ - 2 ) · G ¯ ¯ ( k 2 R ) d s - × × S 2 1 j ω 2 ( - n ¯ 2 × H ¯ - 2 ) · G ¯ ¯ ( k 2 R ) d s + + S 1 ( n ¯ 1 × E ¯ + 2 ) · G ¯ ¯ ( k 2 R ) d s - × × S 1 1 j ω 2 ( n ¯ 1 × H ¯ + 2 ) · G ¯ ¯ ( k 2 R ) d s ; r ¯ { between S 1 and S 2 outside S 2 and inside S 1 .
n ¯ 2 × E ¯ + 3 = n ¯ 2 × E ¯ - 2 n ¯ 2 × H ¯ + 3 = n ¯ 2 × H ¯ - 2 } on S 2 ,
n ¯ 1 × E ¯ + 2 = n ¯ 1 × E ¯ - 1 n ¯ 1 1 × H ¯ + 2 = n ¯ n × H ¯ - 1 } on S 1 .
E ¯ 3 ( r ¯ ) 0 } = E ¯ i ( r ¯ ) + × S 2 ( n ¯ 2 × E ¯ - 2 ) · G ¯ ¯ ( k 3 R ) d s - × × S 2 1 j ω 3 ( n ¯ 2 × H ¯ - 2 ) · G ¯ ¯ ( k 3 R ) d s ; r ¯ { outside S 2 inside S 2 ,
E ¯ 2 ( r ¯ ) 0 } = × S 2 ( - n ¯ 2 × E ¯ - 2 ) · G ¯ ¯ ( k 2 R ) d s - × × S 2 1 k ω 2 ( - n ¯ 2 × H ¯ - 2 ) · G ¯ ¯ ( k 2 R ) d s + × S 1 ( n ¯ 1 × E ¯ - 1 ) · G ¯ ¯ ( k 2 R ) d s - × × S 1 1 j ω 2 ( n ¯ 1 × H ¯ - 1 ) · G ¯ ¯ ( k 2 R ) d s ; r ¯ { between S 1 and S 2 outside S 2 and inside S 1 .
E ¯ i ( r ¯ ) = ν D ν [ a ν M ¯ ν 1 ( k 3 r ¯ ) + b ν N ¯ ν 1 ( k 3 r ¯ ) ] ,
E ¯ s ( r ¯ ) = ν 4 D ν [ f ν M ¯ ν 3 ( k 3 r ¯ ) + g ν N ¯ ν 3 ( k 3 r ¯ ) ] ,
E ¯ 2 ( r ¯ ) = ν D ν [ γ ν M ¯ ν 1 ( k 2 r ¯ ) + δ ν N ¯ ν 1 ( k 2 r ¯ ) + α ν M ¯ ν 3 ( k 2 r ¯ ) + β ν N ¯ ν 3 ( k 2 r ¯ ) ] ,
E ¯ 1 ( r ¯ ) = ν D ν [ c ν M ¯ ν 1 ( k 1 r ¯ ) + d ν N ¯ ν 1 ( k 1 r ¯ ) ] ,
ν = σ = even odd m = 0 n n = 1 ,
[ f g ] = - [ T 2 ] [ a / 4 b / 4 ] ,
[ T 2 ] = { [ Q 2 11 ] - [ Q 2 13 ] [ D ] [ T 1 ] [ D ] - 1 } · { [ Q 2 31 ] - [ Q 2 33 ] [ D ] [ T 1 ] [ D ] - 1 } - 1 .
[ Q i 13 ] = [ K i 13 + ( r i ) 1 / 2 J i 13 L i 13 + ( r i ) 1 / 2 I i 13 I i 13 + ( r i ) 1 / 2 L i 13 J i 13 + ( r i ) 1 / 2 K i 13 ] ,
[ I i 13 ] μ ν = k i + 1 2 π S i n ¯ i · M ¯ μ 1 ( k i + 1 r ¯ ) × M ¯ ν 3 ( k i r ¯ ) d s ,
[ T 1 ] = [ Q 1 11 ] [ Q 1 31 ] - 1 .
[ T n ] = { [ Q n 11 ] - [ Q n 13 ] [ D ] [ T n - 1 ] [ D ] - 1 } · { [ Q n 31 ] - [ Q n 33 ] [ D ] [ T n - 1 ] [ D ] - 1 } - 1 .
E ¯ s ( k r ¯ ) = F ¯ ( θ s , ϕ s / θ i , ϕ i ) exp ( j k r ) r ; k r ,
σ D ( θ s , ϕ s / θ i , ϕ i ) = F ¯ ( θ s , ϕ s / θ i , ϕ i ) 2 .
σ sc = 16 π k 2 ν = 1 N D ν [ f ν 2 + g ν 2 ] ,
σ ext = 4 π k Im [ e ^ 0 · F ¯ ( θ i , ϕ i / θ i , ϕ i ) ] ,
σ abs = σ ext - σ sc ,
σ B = 4 π σ D ( π - θ i , π + ϕ i / θ i , ϕ i ) ,

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