Abstract

A method for studying the scattering properties of a cluster of dielectric spheres is proposed. The vector scattering problem is handled through Debye potentials and a mathematical technique that accounts for multiple scattering effects. The scattered field as well as the scattering and absorption cross sections can be computed without any restriction of principle on the angle of incidence of light and on the radia and refractive indexes of the spheres in the cluster. The resulting expressions take on the well known form when the cluster reduces to a single sphere.

© 1979 Optical Society of America

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References

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  1. G. Mie, Ann. Phys. 25, 377 (1908).
    [CrossRef]
  2. R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Chap. 3.
  3. Rayleigh, Philos. Mag. 41, 107, 274, 447 (1871); Philos. Mag. 12, 81 (1881); G. N. Watson, Proc. R. Soc. London Ser. A: 95, 83 (1918); R. Gans, Ann. Phys. 76, 29 (1925).
    [CrossRef]
  4. M. E. Milham, Edgewood Arsenal Special Publication ED-SP-77002 (June1976); H. Carlon, D. H. Anderson, M. E. Milham, T. L. Tarnove, R. H. Frickel, O. I. Sindoni, Edgewood Arsenal Technical Report ED-TR-77006 (December1976).
  5. P. Debye, Ann. Phys. 30, 57 (1909).
    [CrossRef]
  6. J. C. Slater, Quantum Theory of Molecules and Solids, Vol. 4 (McGraw-Hill, New York, 1974), Chap. 5.
  7. V. Twersky, J. Math. Phys. 8, 589 (1967); W. Trincks, Ann. Phys. 22, 561 (1935); N. Kumagori, D. J. Angelakos, Electron Res. Lab. Rep. 333 (Institute of Engineering Research U. California, Berkeley, 1961); S. Levine, G. O. Olaofe, J. Colloid Interface Sci. 27, 442 (1968); M. Kerker, Ed., Electromagnetic Scattering (Pergamon, Oxford, 1963).
    [CrossRef]
  8. L. Silberstein, Ann. Phys. 22, 24 (1907); Philos. Mag. 23, 790 (1912).
  9. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 16.
  10. R. Nozawa, J. Math. Phys. 7, 1841 (1966); A. R. Williams, S. M. Hu, D. W. Jepsen, in Computational Methods in Band Theory, P. M. Marcus, J. F. Janak, A. R. Williams, Ed. (Plenum, New York, 1971), Appendix IA.
    [CrossRef]
  11. J. A. Gaunt, Proc. Cambridge Philos. Soc. 24, 328 (1928); E. U. Condon, G. H. Shortley, The Theory of Atomic Spectra (Cambridge U. P., Cambridge, 1951), p. 175.
    [CrossRef]
  12. J. C. Slater, Quantum Theory of Atomic Structure, Vol. 1 (McGraw-Hill, New York, 1965), Sec. 13-3.
  13. T. Y. Wu, T. Ohmura, Quantum Theory of Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1962), pp. 15, 317.
  14. J. A. Stratton, Electromagnetic TheoryMcGraw-Hill, New York, 1941), Sec. 57.
  15. M. Kerker, The Scattering of Light (Academic, New York, 1969), p. 250.
  16. J. C. Slater, J. Chem. Phys. 43, S228 (1965).
    [CrossRef]
  17. K. H. Johnson, J. Chem. Phys. 45, 3085 (1966).
    [CrossRef]
  18. K. H. Johnson, F. C. Smith, MIT Scattered Waves Program (unpublished); D. A. Liberman, I. P. Batra, IBM Research Report RJ 1224 (May1973) (unpublished).
  19. F. Herman, A. R. Williams, K. H. Johnson, J. Chem. Phys. 61, 3508 (1974); D. A. Liberman, I. P. Batra, J. Chem. Phys. 59, 3723 (1973).
    [CrossRef]
  20. J. F. Cornwell, Group Theory and Electronic Energy Bands in Solids (North-Holland, Amsterdam, 1969).

1976 (1)

M. E. Milham, Edgewood Arsenal Special Publication ED-SP-77002 (June1976); H. Carlon, D. H. Anderson, M. E. Milham, T. L. Tarnove, R. H. Frickel, O. I. Sindoni, Edgewood Arsenal Technical Report ED-TR-77006 (December1976).

1974 (1)

F. Herman, A. R. Williams, K. H. Johnson, J. Chem. Phys. 61, 3508 (1974); D. A. Liberman, I. P. Batra, J. Chem. Phys. 59, 3723 (1973).
[CrossRef]

1967 (1)

V. Twersky, J. Math. Phys. 8, 589 (1967); W. Trincks, Ann. Phys. 22, 561 (1935); N. Kumagori, D. J. Angelakos, Electron Res. Lab. Rep. 333 (Institute of Engineering Research U. California, Berkeley, 1961); S. Levine, G. O. Olaofe, J. Colloid Interface Sci. 27, 442 (1968); M. Kerker, Ed., Electromagnetic Scattering (Pergamon, Oxford, 1963).
[CrossRef]

1966 (2)

R. Nozawa, J. Math. Phys. 7, 1841 (1966); A. R. Williams, S. M. Hu, D. W. Jepsen, in Computational Methods in Band Theory, P. M. Marcus, J. F. Janak, A. R. Williams, Ed. (Plenum, New York, 1971), Appendix IA.
[CrossRef]

K. H. Johnson, J. Chem. Phys. 45, 3085 (1966).
[CrossRef]

1965 (1)

J. C. Slater, J. Chem. Phys. 43, S228 (1965).
[CrossRef]

1928 (1)

J. A. Gaunt, Proc. Cambridge Philos. Soc. 24, 328 (1928); E. U. Condon, G. H. Shortley, The Theory of Atomic Spectra (Cambridge U. P., Cambridge, 1951), p. 175.
[CrossRef]

1909 (1)

P. Debye, Ann. Phys. 30, 57 (1909).
[CrossRef]

1908 (1)

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

1907 (1)

L. Silberstein, Ann. Phys. 22, 24 (1907); Philos. Mag. 23, 790 (1912).

1871 (1)

Rayleigh, Philos. Mag. 41, 107, 274, 447 (1871); Philos. Mag. 12, 81 (1881); G. N. Watson, Proc. R. Soc. London Ser. A: 95, 83 (1918); R. Gans, Ann. Phys. 76, 29 (1925).
[CrossRef]

Cornwell, J. F.

J. F. Cornwell, Group Theory and Electronic Energy Bands in Solids (North-Holland, Amsterdam, 1969).

Debye, P.

P. Debye, Ann. Phys. 30, 57 (1909).
[CrossRef]

Gaunt, J. A.

J. A. Gaunt, Proc. Cambridge Philos. Soc. 24, 328 (1928); E. U. Condon, G. H. Shortley, The Theory of Atomic Spectra (Cambridge U. P., Cambridge, 1951), p. 175.
[CrossRef]

Herman, F.

F. Herman, A. R. Williams, K. H. Johnson, J. Chem. Phys. 61, 3508 (1974); D. A. Liberman, I. P. Batra, J. Chem. Phys. 59, 3723 (1973).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 16.

Johnson, K. H.

F. Herman, A. R. Williams, K. H. Johnson, J. Chem. Phys. 61, 3508 (1974); D. A. Liberman, I. P. Batra, J. Chem. Phys. 59, 3723 (1973).
[CrossRef]

K. H. Johnson, J. Chem. Phys. 45, 3085 (1966).
[CrossRef]

K. H. Johnson, F. C. Smith, MIT Scattered Waves Program (unpublished); D. A. Liberman, I. P. Batra, IBM Research Report RJ 1224 (May1973) (unpublished).

Kerker, M.

M. Kerker, The Scattering of Light (Academic, New York, 1969), p. 250.

Mie, G.

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Milham, M. E.

M. E. Milham, Edgewood Arsenal Special Publication ED-SP-77002 (June1976); H. Carlon, D. H. Anderson, M. E. Milham, T. L. Tarnove, R. H. Frickel, O. I. Sindoni, Edgewood Arsenal Technical Report ED-TR-77006 (December1976).

Newton, R. G.

R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Chap. 3.

Nozawa, R.

R. Nozawa, J. Math. Phys. 7, 1841 (1966); A. R. Williams, S. M. Hu, D. W. Jepsen, in Computational Methods in Band Theory, P. M. Marcus, J. F. Janak, A. R. Williams, Ed. (Plenum, New York, 1971), Appendix IA.
[CrossRef]

Ohmura, T.

T. Y. Wu, T. Ohmura, Quantum Theory of Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1962), pp. 15, 317.

Rayleigh,

Rayleigh, Philos. Mag. 41, 107, 274, 447 (1871); Philos. Mag. 12, 81 (1881); G. N. Watson, Proc. R. Soc. London Ser. A: 95, 83 (1918); R. Gans, Ann. Phys. 76, 29 (1925).
[CrossRef]

Silberstein, L.

L. Silberstein, Ann. Phys. 22, 24 (1907); Philos. Mag. 23, 790 (1912).

Slater, J. C.

J. C. Slater, J. Chem. Phys. 43, S228 (1965).
[CrossRef]

J. C. Slater, Quantum Theory of Atomic Structure, Vol. 1 (McGraw-Hill, New York, 1965), Sec. 13-3.

J. C. Slater, Quantum Theory of Molecules and Solids, Vol. 4 (McGraw-Hill, New York, 1974), Chap. 5.

Smith, F. C.

K. H. Johnson, F. C. Smith, MIT Scattered Waves Program (unpublished); D. A. Liberman, I. P. Batra, IBM Research Report RJ 1224 (May1973) (unpublished).

Stratton, J. A.

J. A. Stratton, Electromagnetic TheoryMcGraw-Hill, New York, 1941), Sec. 57.

Twersky, V.

V. Twersky, J. Math. Phys. 8, 589 (1967); W. Trincks, Ann. Phys. 22, 561 (1935); N. Kumagori, D. J. Angelakos, Electron Res. Lab. Rep. 333 (Institute of Engineering Research U. California, Berkeley, 1961); S. Levine, G. O. Olaofe, J. Colloid Interface Sci. 27, 442 (1968); M. Kerker, Ed., Electromagnetic Scattering (Pergamon, Oxford, 1963).
[CrossRef]

Williams, A. R.

F. Herman, A. R. Williams, K. H. Johnson, J. Chem. Phys. 61, 3508 (1974); D. A. Liberman, I. P. Batra, J. Chem. Phys. 59, 3723 (1973).
[CrossRef]

Wu, T. Y.

T. Y. Wu, T. Ohmura, Quantum Theory of Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1962), pp. 15, 317.

Ann. Phys. (3)

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

P. Debye, Ann. Phys. 30, 57 (1909).
[CrossRef]

L. Silberstein, Ann. Phys. 22, 24 (1907); Philos. Mag. 23, 790 (1912).

Edgewood Arsenal Special Publication ED-SP-77002 (1)

M. E. Milham, Edgewood Arsenal Special Publication ED-SP-77002 (June1976); H. Carlon, D. H. Anderson, M. E. Milham, T. L. Tarnove, R. H. Frickel, O. I. Sindoni, Edgewood Arsenal Technical Report ED-TR-77006 (December1976).

J. Chem. Phys. (3)

J. C. Slater, J. Chem. Phys. 43, S228 (1965).
[CrossRef]

K. H. Johnson, J. Chem. Phys. 45, 3085 (1966).
[CrossRef]

F. Herman, A. R. Williams, K. H. Johnson, J. Chem. Phys. 61, 3508 (1974); D. A. Liberman, I. P. Batra, J. Chem. Phys. 59, 3723 (1973).
[CrossRef]

J. Math. Phys. (2)

R. Nozawa, J. Math. Phys. 7, 1841 (1966); A. R. Williams, S. M. Hu, D. W. Jepsen, in Computational Methods in Band Theory, P. M. Marcus, J. F. Janak, A. R. Williams, Ed. (Plenum, New York, 1971), Appendix IA.
[CrossRef]

V. Twersky, J. Math. Phys. 8, 589 (1967); W. Trincks, Ann. Phys. 22, 561 (1935); N. Kumagori, D. J. Angelakos, Electron Res. Lab. Rep. 333 (Institute of Engineering Research U. California, Berkeley, 1961); S. Levine, G. O. Olaofe, J. Colloid Interface Sci. 27, 442 (1968); M. Kerker, Ed., Electromagnetic Scattering (Pergamon, Oxford, 1963).
[CrossRef]

Philos. Mag. (1)

Rayleigh, Philos. Mag. 41, 107, 274, 447 (1871); Philos. Mag. 12, 81 (1881); G. N. Watson, Proc. R. Soc. London Ser. A: 95, 83 (1918); R. Gans, Ann. Phys. 76, 29 (1925).
[CrossRef]

Proc. Cambridge Philos. Soc. (1)

J. A. Gaunt, Proc. Cambridge Philos. Soc. 24, 328 (1928); E. U. Condon, G. H. Shortley, The Theory of Atomic Spectra (Cambridge U. P., Cambridge, 1951), p. 175.
[CrossRef]

Other (9)

J. C. Slater, Quantum Theory of Atomic Structure, Vol. 1 (McGraw-Hill, New York, 1965), Sec. 13-3.

T. Y. Wu, T. Ohmura, Quantum Theory of Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1962), pp. 15, 317.

J. A. Stratton, Electromagnetic TheoryMcGraw-Hill, New York, 1941), Sec. 57.

M. Kerker, The Scattering of Light (Academic, New York, 1969), p. 250.

J. F. Cornwell, Group Theory and Electronic Energy Bands in Solids (North-Holland, Amsterdam, 1969).

K. H. Johnson, F. C. Smith, MIT Scattered Waves Program (unpublished); D. A. Liberman, I. P. Batra, IBM Research Report RJ 1224 (May1973) (unpublished).

R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Chap. 3.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 16.

J. C. Slater, Quantum Theory of Molecules and Solids, Vol. 4 (McGraw-Hill, New York, 1974), Chap. 5.

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Equations (41)

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× F ± = ± k n F ± ,
( 2 + k 2 n 2 ) ϕ = 0 ,             ( ϕ 2 + k 2 n 2 ) ψ = 0.
E = k L ψ + [ 1 / ( n 2 ) ] × L ϕ ,
B = - i k L ϕ - i × L ψ ,
E ± = ( e x ± e y ) E 0 exp ( i k z ) = E 0 l i l d l [ L j l ( k r ) Y l , ± 1 ( r ^ ) ± 1 k × L j l ( k r ) Y l , ± 1 ( r ^ ) ]
B ± = i E ± ,
d l = [ 4 π 2 l + 1 l ( l + 1 ) ] 1 / 2 .
ψ 0 ± = E 0 1 k l i l d l j l ( k r ) Y l , ± 1 ( r ^ ) ,
ϕ 0 ± = ± E 0 1 k l i l d l j l ( k r ) Y l , ± 1 ( r ^ ) .
n ( r ) = n ( r a + R α ) = { n α             : r α b α 1             : r α > b a ,
ϕ 1 ( r α ) = ± E 0 2 k l m i l d l C l m α Y l , m ( r ^ α ) j l ( K α r α ) ,
ψ 1 ( r α ) = E 0 2 k l m i l d l D l m α Y l , m ( r ^ α ) j l ( K α r α ) .
ϕ II ( r ) = ϕ 0 ± ( r ) ± E 0 2 k α l m i l d l A l m α Y l m ( r ^ α ) h l ( 1 ) ( k r α ) ,
ψ II ( r ) = ψ 0 ± ( r ) ± E 0 2 k α l m i l d l B l m α Y l m ( r ^ α ) h l ( 1 ) ( k r α ) .
ϕ 0 ± ( r ) = ϕ 0 ± ( r α + R α ) = exp ( i k · R α ) ϕ 0 ± ( r α ) ,
ψ 0 ± ( r ) = ψ 0 ± ( r α + R α ) = exp ( i k · R α ) ψ 0 ± ( r α ) ,
Y l m ( r ^ β ) h l ( 1 ) ( k r β ) = 4 π L l m I L ( l m ; l m ) i l - l - L · Y L , m - m * ( R ^ α β ) h L ( 1 ) ( kR α β ) Y l m ( r ^ α ) j l ( k r α ) ,
I L ( l m ; l m ) = Y L , m - m ( r ^ ) Y l m * ( r ^ ) Y l m ( r ^ ) d Ω
ϕ II ( r α ) = ± E 0 2 k l m i l d l Y l m ( r ^ α ) [ 2 exp ( i k · R α ) j l ( k r α ) δ m , ± 1 + A l m α h l ( 1 ) ( k r α ) + β l m G l m ; l m α β ( k ) A l m β j l ( k r α ) ] ,
ψ II ( r α ) = E 0 2 k l m i l d l Y l m ( r ^ α ) [ 2 exp ( i k · R α ) j l ( k r α ) δ m , ± 1 + B l m α h l ( 1 ) ( k r α ) + β l m G l m ; l m α β ( k ) B l m β j l ( k r α ) ] ,
G l m ; l m α β ( k ) = 4 π ( 1 - δ α β ) d l d l × L I L ( l m ; l m ) i - L Y L , m - m * ( R ^ α β ) h L ( 1 ) ( k R α β ) .
[ ϕ I ( r α ) ] r α = b α = [ ϕ II ( r α ) ] r α = b α , 1 n a 2 [ r α r α ϕ I ( r α ) ] r α = b α = [ r α r α ϕ II ( r α ) ] r α = b α , [ ψ I ( r α ) ] r α = b α = [ ψ II ( r α ) ] r α = b α , [ r α r α ψ I ( r α ) ] r α = b α = [ r α r α ψ II ( r α ) ] r α = b α ,
l m i l d l Y l m ( r ^ α ) [ 2 exp ( i k · R α ) j l ( k r α ) δ m , ± 1 + A l m α h l ( 1 ) ( k r α ) + β l m G l m ; l m α β A l , m β j l ( k r α ) ] r α = b α = [ l m i l d l C l m α Y l m ( r ^ α ) j l ( k r α ) ] r α = b α ,
l m i l d l C l m α Y l m ( r ^ α ) 1 n α 2 [ d d r α r α j l ( K α r α ) ] r α = b α = l m i l d l Y l m ( r ^ α ) { 2 exp ( i k · R α ) [ d d r α r α j l ( k r α ) ] δ m , ± 1 + A l m α [ d d r α r α h l ( 1 ) ( k r α ) ] + β l m G l m ; l m α β A l m β [ d d r α r α j l ( k r α ) ] } r α = b α ,
l m i l d l Y l m ( r ^ α ) [ 2 exp ( i k · R α ) j l ( k r α ) δ m , ± 1 + B l m α h l ( 1 ) ( k r α ) + β l m G l m ; l m α β B l m β j l ( k r α ) ] r α = b α = [ l m i l d l D l m α Y l m ( r ^ α ) j l ( k r α ) ] r α = b α ,
l m i l d l D l m α Y l m ( r ^ α ) [ d d r α r α j l ( K α r α ) ] r α = b α = l m i l d l Y l m ( r ^ α ) { 2 exp ( i k · R α ) · [ d d r α r α j l ( k r α ) ] δ m , ± 1 + B l m α [ d d r α r α h l ( 1 ) ( k r α ) ] + β l m G l m ; l m α β B l m β [ d d r α r α j l ( k r α ) ] } r α = b α .
β l m { δ α β δ l l δ m m [ S l β ( K β ) ] - 1 + G l m ; l m α β } A l m β = - 2 exp ( i k · R α ) δ m , ± 1 ,
β l m { δ α β δ l l δ m m [ T l β ( K β ) ] - 1 + G l m ; l m α β } B l m β = - 2 exp ( i k · R α ) δ m , ± 1 .
S l α ( K α ) = { j l ( K α r α ) [ d d r α r α j l ( k r α ) ] - 1 n α 2 j l ( k r α ) [ d d r α r α j l ( K α r α ) ] j l ( K α r α ) [ d d r α r α h l ( 1 ) ( k r α ) ] - 1 n α 2 h l ( 1 ) ( k r α ) [ d d r α r α j l ( K α r α ) ] } r α = b α ,
T l α ( K α ) = { j l ( K α r α ) [ d d r α r α j l ( k r α ) ] - j l ( k r α ) [ d d r α r α j l ( K α r α ) ] j l ( K α r α ) [ d d r α r α h l ( 1 ) ( k r α ) ] - h l ( 1 ) ( k r α ) [ d d r α r α j l ( K α r α ) ] } r α = b α ,
C l m α = { h l ( 1 ) ( k r α ) [ d d r α r α j l ( k r α ) ] - j l ( k r α ) [ d d r α r α h l ( 1 ) ( k r α ) ] j l ( K α r α ) [ d d r α r α j l ( k r α ) ] - 1 n α 2 j l ( k r α ) [ d d r α r α j l ( K α r α ) ] } r α = b α A l m α ,
D l m α = { h l ( 1 ) ( k r α ) [ d d r α r α j l ( k r α ) ] - j l ( k r α ) [ d d r α r α h l ( 1 ) ( k r α ) ] j l ( K α r α ) [ d d r α r α j l ( k r α ) ] - j l ( k r α ) [ d d r α r α j l ( K α r α ) ] } r α = b α B l m α ,
ϕ ( r ) ~ ϕ 0 ± ( r ) ± E 0 2 k α l m i l d l A l m α Y l m ( r ^ ) h l ( 1 ) ( k r ) , ψ ( r ) ~ ψ 0 ± + E 0 2 k α l m i l d l B l m α Y l m ( r ^ ) h l ( 1 ) ( k r ) , }
E s = E 0 2 α l m i l [ 4 π ( 2 l + 1 ) ] 1 / 2 [ B l m α h l ( 1 ) ( k r ) X l m ( r ^ ) ± A l m α × h l ( 1 ) ( k r ) X l m ( r ^ ) ] , B s = - i E 0 2 α l m i l [ 4 π ( 2 l + 1 ) ] 1 / 2 [ ± A l m α h l ( 1 ) ( k r ) X l m ( r ^ ) + B l m α × h l ( 1 ) ( k r ) X l m ( r ^ ) ] .
σ s = π 2 k 2 α β l m ( 2 l + 1 ) [ A l m α A l m β * + B l m α B l m β * ] ,
σ abs = π 2 k 2 α β l m ( 2 l + 1 ) [ 2 - ( A l m α - 1 ) ( A l m β * - 1 ) - ( β l m α - 1 ) ( B l m β * - 1 ) ] .
σ t = - π 2 k 2 α β l m ( 2 l + 1 ) [ A l m α + A l m β * + B l m α + B l m β * ] .
G l m ; l m α α = 0.
A l m = - 2 δ m , ± 1 S l ( K ) ,
B l m = - 2 δ m , ± 1 T l ( K ) .
σ t = - π k 2 l ( 2 l + 1 ) Re ( A l + B l )

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