Abstract

The Rayleigh-Gans approximation theory is applied to the problem of scattering by two neighboring spherical particles. For simplicity, we have examined in detail the case of two identical uniform spheres. Differential intensities for any arbitrary location of the particle pair relative to the incident wave, mean intensities, and mean scattering cross sections for randomly oriented particles are considered. The results show good agreement with the exact theory. Some numerical results are presented for the particular case of touching spheres (dumbbells), which are either at random, or are at specific, orientation relative to the direction of polarization of the incident wave. It is observed that for small spheres at random orientation, the scattering cross section is four times the value for that of a single sphere. This factor is two for large particles. We also observe that both the intensity and dissymmetry patterns for two spheres are totally different from the single particle ones.

© 1970 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Mie, Ann. Physik 25, 377 (1908).
    [CrossRef]
  2. G. O. Olaofe, “Scattering of Electromagnetic Waves by Spherical Particles,” Ph.D. Thesis, University of Manchester, 1965.
  3. W. Trinks, Ann. Physik 22, 561 (1935).
    [CrossRef]
  4. O. A. Germogenova, Izv. Akad. Nauk. USSR Geofiz. 4, 648 (1963) [Bull. Acad. Sci. USSR, Geophys. Ser. (USA) 4, 403 (1963)].
  5. S. Levine, G. O. Olaofe, J. Colloid Interface Sci. 27, 442 (1968).
    [CrossRef]
  6. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, New York, 1966).
  7. H. Benoit, R. Ullman, A. J. De Vries, C. Wippler, J. Chim. Phys. 59, 889 (1962).
  8. Rayleigh, Proc. Roy. Soc. London A90, 219 (1914).
  9. H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).
  10. P. Doty, R. F. Steiner, J. Chem. Phys. 18, 1211 (1950).
    [CrossRef]
  11. D. H. Napper, R. H. Ottewill, Kolloid-Z. 192, 114 (1963).
    [CrossRef]
  12. M. Kerker, J. P. Kratohvil, E. Matijevic, J. Opt. Soc. Amer. 52, 551 (1962).
    [CrossRef]
  13. G. O. Olaofe, S. Levine, Electromagnetic Scattering, Proceedings ICES II, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1965), pp. 237–292.

1968 (1)

S. Levine, G. O. Olaofe, J. Colloid Interface Sci. 27, 442 (1968).
[CrossRef]

1963 (2)

O. A. Germogenova, Izv. Akad. Nauk. USSR Geofiz. 4, 648 (1963) [Bull. Acad. Sci. USSR, Geophys. Ser. (USA) 4, 403 (1963)].

D. H. Napper, R. H. Ottewill, Kolloid-Z. 192, 114 (1963).
[CrossRef]

1962 (2)

M. Kerker, J. P. Kratohvil, E. Matijevic, J. Opt. Soc. Amer. 52, 551 (1962).
[CrossRef]

H. Benoit, R. Ullman, A. J. De Vries, C. Wippler, J. Chim. Phys. 59, 889 (1962).

1950 (1)

P. Doty, R. F. Steiner, J. Chem. Phys. 18, 1211 (1950).
[CrossRef]

1935 (1)

W. Trinks, Ann. Physik 22, 561 (1935).
[CrossRef]

1914 (1)

Rayleigh, Proc. Roy. Soc. London A90, 219 (1914).

1908 (1)

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

Benoit, H.

H. Benoit, R. Ullman, A. J. De Vries, C. Wippler, J. Chim. Phys. 59, 889 (1962).

De Vries, A. J.

H. Benoit, R. Ullman, A. J. De Vries, C. Wippler, J. Chim. Phys. 59, 889 (1962).

Doty, P.

P. Doty, R. F. Steiner, J. Chem. Phys. 18, 1211 (1950).
[CrossRef]

Germogenova, O. A.

O. A. Germogenova, Izv. Akad. Nauk. USSR Geofiz. 4, 648 (1963) [Bull. Acad. Sci. USSR, Geophys. Ser. (USA) 4, 403 (1963)].

Kerker, M.

M. Kerker, J. P. Kratohvil, E. Matijevic, J. Opt. Soc. Amer. 52, 551 (1962).
[CrossRef]

Kratohvil, J. P.

M. Kerker, J. P. Kratohvil, E. Matijevic, J. Opt. Soc. Amer. 52, 551 (1962).
[CrossRef]

Levine, S.

S. Levine, G. O. Olaofe, J. Colloid Interface Sci. 27, 442 (1968).
[CrossRef]

G. O. Olaofe, S. Levine, Electromagnetic Scattering, Proceedings ICES II, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1965), pp. 237–292.

Matijevic, E.

M. Kerker, J. P. Kratohvil, E. Matijevic, J. Opt. Soc. Amer. 52, 551 (1962).
[CrossRef]

Mie, G.

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

Napper, D. H.

D. H. Napper, R. H. Ottewill, Kolloid-Z. 192, 114 (1963).
[CrossRef]

Olaofe, G. O.

S. Levine, G. O. Olaofe, J. Colloid Interface Sci. 27, 442 (1968).
[CrossRef]

G. O. Olaofe, “Scattering of Electromagnetic Waves by Spherical Particles,” Ph.D. Thesis, University of Manchester, 1965.

G. O. Olaofe, S. Levine, Electromagnetic Scattering, Proceedings ICES II, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1965), pp. 237–292.

Ottewill, R. H.

D. H. Napper, R. H. Ottewill, Kolloid-Z. 192, 114 (1963).
[CrossRef]

Rayleigh,

Rayleigh, Proc. Roy. Soc. London A90, 219 (1914).

Steiner, R. F.

P. Doty, R. F. Steiner, J. Chem. Phys. 18, 1211 (1950).
[CrossRef]

Trinks, W.

W. Trinks, Ann. Physik 22, 561 (1935).
[CrossRef]

Ullman, R.

H. Benoit, R. Ullman, A. J. De Vries, C. Wippler, J. Chim. Phys. 59, 889 (1962).

van de Hulst, H. C.

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

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, New York, 1966).

Wippler, C.

H. Benoit, R. Ullman, A. J. De Vries, C. Wippler, J. Chim. Phys. 59, 889 (1962).

Ann. Physik (2)

W. Trinks, Ann. Physik 22, 561 (1935).
[CrossRef]

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

Izv. Akad. Nauk. USSR Geofiz. (1)

O. A. Germogenova, Izv. Akad. Nauk. USSR Geofiz. 4, 648 (1963) [Bull. Acad. Sci. USSR, Geophys. Ser. (USA) 4, 403 (1963)].

J. Chem. Phys. (1)

P. Doty, R. F. Steiner, J. Chem. Phys. 18, 1211 (1950).
[CrossRef]

J. Chim. Phys. (1)

H. Benoit, R. Ullman, A. J. De Vries, C. Wippler, J. Chim. Phys. 59, 889 (1962).

J. Colloid Interface Sci. (1)

S. Levine, G. O. Olaofe, J. Colloid Interface Sci. 27, 442 (1968).
[CrossRef]

J. Opt. Soc. Amer. (1)

M. Kerker, J. P. Kratohvil, E. Matijevic, J. Opt. Soc. Amer. 52, 551 (1962).
[CrossRef]

Kolloid-Z. (1)

D. H. Napper, R. H. Ottewill, Kolloid-Z. 192, 114 (1963).
[CrossRef]

Proc. Roy. Soc. London (1)

Rayleigh, Proc. Roy. Soc. London A90, 219 (1914).

Other (4)

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

G. O. Olaofe, “Scattering of Electromagnetic Waves by Spherical Particles,” Ph.D. Thesis, University of Manchester, 1965.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, New York, 1966).

G. O. Olaofe, S. Levine, Electromagnetic Scattering, Proceedings ICES II, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1965), pp. 237–292.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

The geometry of the problem.

Fig. 2
Fig. 2

Z45-Dissymmetry.

Fig. 3
Fig. 3

Multiple scattering intensity by dumbbells (x = 0.5 and m = 1.05).

Fig. 4
Fig. 4

Multiple scattering intensity by dumbbells (x = 0.5 and m = 1.1).

Fig. 5
Fig. 5

Multiple scattering intensity by dumbbells (x = 0.5 and m = 1.2).

Fig. 6
Fig. 6

Multiple scattering intensity by dumbbells (x = 1 and m = 1.05).

Fig. 7
Fig. 7

Effect of separation on multiple scattering by end illuminated dumbbells (x = 1 and m = 1.1).

Fig. 8
Fig. 8

Effect of separation on multiple scattering by broadside illuminated dumbbells (Case 1, β = 0) (x = 1 and m = 1.1).

Fig. 9
Fig. 9

Effect of separation on multiple scattering by broadside illuminated dumbbells (Case 2, β = π/2) (x = 1 and m = 1.1).

Tables (6)

Tables Icon

Table I Dissymmetry … Z45

Tables Icon

Table II Comparison of Differential Intensities of Scattering from Randomly Oriented Spheres with the Corresponding Results for Suitably Chosen Concentric Spheres

Tables Icon

Table III Comparison of Differential Intensities of Scattering from Randomly Oriented Spheres with the Corresponding Results for Suitably Chosen Concentric Spheres

Tables Icon

Table IV Comparison of Differential Intensities of Scattering from Randomly Oriented Spheres with the Corresponding Results for Suitably Chosen Concentric Spheres

Tables Icon

Table V Total Scattering Cross Section

Tables Icon

Table VI Effect of Separation on the Total Scattering Coefficient

Equations (57)

Equations on this page are rendered with MathJax. Learn more.

| n 1 | 1 ; 2 x | n 1 | 1 ,
E s c a = ( i x 3 / k r ) S ( γ,λ ) [ e x ( e x · e r ) e r ] .
S ( γ,λ ) = V α ( r ) exp [ i k r · ( e r e z ) ] d V ( r ) ,
S ( γ,λ ) = V α ( r ) exp [ i k ( r + c / k ) · ( e r e z ) ] d V ( r ) ,
f ( u ) = ( sin u u cos u ) / u 3 ,
I 1 = s 2 | E λ | 2 = i 1 sin 2 λ,
I 2 = s 2 | E γ | 2 = i 2 cos 2 λ
I 1 = 2 sin 2 λ x 6 ( m 2 1 ) 2 f 2 ( u ) g ( α , β , γ , λ ) ,
I 2 = 2 cos 2 π cos 2 λ x 6 ( m 2 1 ) 2 f 2 ( u ) g ( α , β , γ , λ ) ,
g ( α , β , γ , λ ) = 1 + cos [ c · ( e x e r ) ] .
cos [ c · ( e x e r ) ] = cos [ c cos α ( 1 cos γ ) c sin α sin γ cos ( λ β ) ] .
μ = c cos α ( 1 cos γ ) and ν = c sin α sin γ,
g ( α , β , γ , λ ) = 1 + cos μ cos [ ν cos ( λ β ) ] + sin μ sin [ ν cos ( λ β ) ]
i 1 = 2 x 6 ( m 2 1 ) 2 f 2 ( u ) { 1 + cos μ cos [ ν cos ( λ β ) ] + sin μ sin [ ν cos ( λ β ) ] } ,
i 2 = i 1 cos 2 γ .
i 1 = i 1 ( 0 ) = x 6 ( m 2 1 ) 2 f 2 ( u ) ; i 2 = i 2 ( 0 ) = i 1 ( 0 ) cos 2 γ .
α = 0 , and α = π / 2 .
i 1 = i 10 = 2 x 6 ( m 2 1 ) 2 f 2 ( u ) [ 1 + cos ( 1 cos γ ) ] ,
i 2 = i 20 = i 10 cos 2 γ .
i 1 = i 11 = 2 x 6 ( m 2 1 ) 2 f 2 ( u ) { 1 + cos [ c sin γ cos ( λ β ) ] } , i 2 = i 21 = i 11 cos 2 γ .
C s c a ( α , β ) = 0 2 π 0 π ( i 1 sin 2 λ + i 2 cos 2 λ ) sin γ d γ d λ .
cos ( z cos θ ) = J 0 ( z ) + 2 n = 1 ( 1 ) n J 2 n ( z ) cos 2 n θ
sin ( z cos θ ) = 2 n = 0 ( 1 ) n J 2 n + 1 ( z ) cos ( 2 n + 1 ) θ .
0 π cos sin [ ν cos ( λ β ) ] d λ = { 2 π J 0 ( ν ) 0
0 π cos 2 λ cos sin [ ν cos ( λ β ) ] d λ = { π 0 [ J 0 ( ν ) J 2 ( ν ) cos 2 β ] .
C s c a ( α , β ) = 2 π x 6 ( m 2 1 ) 0 π { ( 1 + cos 2 γ ) [ 1 + cos μ J 0 ( ν ) + J 2 ( ν ) cos μ sin 2 γ cos 2 β ] } sin γ d γ .
i 1 β = 2 x 6 ( m 2 1 ) 2 f 2 ( u ) 0 2 π { 1 + cos μ cos [ ν cos ( λ β ) ] + sin μ sin [ ν cos ( λ β ) ] } d β 2 π .
i 1 β = 2 x 6 ( m 2 1 ) 2 f 2 ( u ) { 1 + cos [ c cos α × ( 1 cos γ ) ] J 0 ( c sin α sin γ ) } .
1 2 0 π cos [ c cos α ( 1 cos γ ) ] J 0 ( c sin α sin γ ) sin γ d γ .
0 π cos ( z cos θ cos ϕ ) J υ 1 2 ( z sin θ sin ϕ ) C υ r ( cos θ ) sin υ + 1 2 θ d θ = ( ) r / 2 ( 2 π / z ) 1 2 sin υ 1 2 ϕ C υ r ( cos ϕ ) J υ + r ( z ) , if r is even = 0 , if r is odd .
i 1 β α = 2 x 6 ( m 2 1 ) 2 f 2 ( u ) [ 1 + ( sin η u / η u ) ] .
C s c a β α = 2 π ( m 2 1 ) 2 x 6 0 π ( 1 + sin η u η u ) f 2 ( u ) × sin γ ( 1 + cos 2 γ ) d γ .
g = 1 2 j = 1 N j = 1 N cos [ c j j · ( e x e r ) ] ,
j = 1 N j = 1 N sin η j j u η j j u ,
C s c a β α c = 2 π ( m 2 1 ) 2 x 6 0 π f 2 ( u ) ( 1 + cos 2 γ ) sin γ d γ
G = 0 π / 2 G sin α d α ,
cos w = c sin α / 2 a ,
G = a 2 ( 2 π 2 w + sin 2 w ) ,
G = a 2 ( π + 2 α + sin 2 α ) .
Q s c a β α = 2 d ( m 2 1 ) 2 x 4 0 π f 2 ( u ) [ 1 + cos 2 ( γ ) ] ( 1 + sin 2 u 2 u ) × sin γ d γ .
C s c a β α = ( 32 / 27 ) π ( m 2 1 ) 2 x 6 ,
C s c a β α = π ( m 2 1 ) 2 x 4 ,
[ E γ E λ ] = [ S 2 S 3 S 4 S 1 ] [ E o l E o r ] .
i 1 ( = i ) = I 1 [ λ = ( π / 2 ) ] .
i 2 ( = i ) = I 2 ( λ = 0 ) .
i 1 = S 0 cos 2 ( c sin 2 1 2 γ ) ,
i 2 = i 1 cos 2 γ ,
S 0 = 4 x 6 ( m 2 1 ) 2 f 2 ( u ) .
1 3 2 c 1 [ ζ 1 ( c ) / c ] e i c 1 9 4 c 1 2 [ ζ 1 ( c ) / c 2 ]
e i c 3 2 c , [ ζ 1 ( c ) / c ] 1 9 4 c 1 2 [ ζ 1 ( c ) / c 2 ] ,
ζ 1 ( c ) = ( π c / 2 ) 1 2 H 3 2 ( 1 ) ( c ) ,
1 + 0 ( n 2 1 ) ( x 3 / c 3 ) .
i 1 = S 0 , i 2 = S 0 cos 2 ( 1 2 c sin γ ) cos 2 γ,
i 1 = S 0 cos 2 ( 1 2 c sin γ ) , i 2 = S 0 cos 2 γ .
i 1 = 1 4 S 0 , i 2 = 1 4 S 0 cos 2 γ,
i 2 = i 1 cos 2 γ
Z 45 = i 45 / i 35 .

Metrics