Abstract

Each spike that is observed in the backscattering from large dielectric spheres arises from just one term in the Mie series. By approximating the Ricatti–Bessel functions involved, one obtains expressions for the location and width of each spike. Every single spike can be allocated to just two patterns, which are described in terms of sequences of two integers and whose repetition period is a function only of the refractive index m. The physical process involved is shown to be a resonance inside the sphere of the electric or magnetic field, in which the appropriate field increases rapidly to a large value at a radial distance r/a ~ 1/m, whereas the other field remains small. As the sphere surface is approached, they combine to form a simple, spherical electromagnetic standing-wave pattern of large amplitude.

© 1984 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Mie, “Beitrage zur Optik truber Median, speziell kolloider Metallosungen,” Ann. Phys. 25, 377–445 (1908).
    [CrossRef]
  2. D. Atlas, K. M. Glover, “Back-scatter by dielectric spheres with and without metal caps,” in Interdisciplinary Conference on Electromagnetic Scattering, M. Kerker, ed. (Pergamon, New York, 1963), pp. 213–236.
  3. H. C. Bryant, A. J. Cox, “Mie theory and the glory,” J. Opt. Soc. Am. 56, 1529–1532 (1966).
    [CrossRef]
  4. T. S. Fahlen, H. C. Bryant, “Optical backscattering from single water drops,” J. Opt. Soc. Am. 58, 304–310 (1968).
    [CrossRef]
  5. H. C. van de Hulst, in Light Scattering by Small Particles (Dover, New York, 1981), pp. 373–374.
  6. J. R. Probert-Jones, “Surface waves associated with the backscattering of microwave radiation by large ice spheres,” in Interdisciplinary Conference on Electromagnetic Scattering, M. Kerker, ed. (Pergamon, New York, 1963), pp. 237–250.
  7. B. M. Herman, L. J. Battan, “Calculations of Mie backscattering of microwaves from ice spheres,” Quart. J. R. Meteorol. Soc. 87, 223–230 (1961).
    [CrossRef]
  8. J. R. Probert-Jones, “Surface waves in backscattering and the localization principle,” J. Opt. Soc. Am. 73, 503 (1983).
    [CrossRef]
  9. P. Chylek, J. T. Kiehl, M. K. W. Ko, A. Ashkin, “Surface waves in light scattering by spherical and non-spherical particles,” in Light Scattering by Irregularly Shaped Particles, D. W. Shuerman, ed. (Plenum, New York, 1980), pp. 153–164.
    [CrossRef]
  10. H. C. van de Hulst, in Light Scattering by Small Particles (Dover, New York, 1981), p. 135.
  11. M. Kerker, in The Scattering of Light and Other Electromagnetic Radiation (Academic Press, New York, 1969), p. 68.
  12. P. M. Morse, H. Feshbach, in Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 631.
  13. H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979).
    [CrossRef]
  14. H. C. van de Hulst, in Light Scattering by Small Particles (Dover, New York, 1981), pp. 155–158.
  15. J. F. Owen, R. K. Chang, P. W. Barber, “Internal electric field distributions of a dielectric cylinder at resonance wavelengths,” Opt. Lett. 6, 540–542 (1981).
    [CrossRef] [PubMed]

1983 (1)

1981 (1)

1979 (1)

1968 (1)

1966 (1)

1961 (1)

B. M. Herman, L. J. Battan, “Calculations of Mie backscattering of microwaves from ice spheres,” Quart. J. R. Meteorol. Soc. 87, 223–230 (1961).
[CrossRef]

1908 (1)

G. Mie, “Beitrage zur Optik truber Median, speziell kolloider Metallosungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Ashkin, A.

P. Chylek, J. T. Kiehl, M. K. W. Ko, A. Ashkin, “Surface waves in light scattering by spherical and non-spherical particles,” in Light Scattering by Irregularly Shaped Particles, D. W. Shuerman, ed. (Plenum, New York, 1980), pp. 153–164.
[CrossRef]

Atlas, D.

D. Atlas, K. M. Glover, “Back-scatter by dielectric spheres with and without metal caps,” in Interdisciplinary Conference on Electromagnetic Scattering, M. Kerker, ed. (Pergamon, New York, 1963), pp. 213–236.

Barber, P. W.

Battan, L. J.

B. M. Herman, L. J. Battan, “Calculations of Mie backscattering of microwaves from ice spheres,” Quart. J. R. Meteorol. Soc. 87, 223–230 (1961).
[CrossRef]

Bryant, H. C.

Chang, R. K.

Chylek, P.

P. Chylek, J. T. Kiehl, M. K. W. Ko, A. Ashkin, “Surface waves in light scattering by spherical and non-spherical particles,” in Light Scattering by Irregularly Shaped Particles, D. W. Shuerman, ed. (Plenum, New York, 1980), pp. 153–164.
[CrossRef]

Cox, A. J.

Fahlen, T. S.

Feshbach, H.

P. M. Morse, H. Feshbach, in Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 631.

Glover, K. M.

D. Atlas, K. M. Glover, “Back-scatter by dielectric spheres with and without metal caps,” in Interdisciplinary Conference on Electromagnetic Scattering, M. Kerker, ed. (Pergamon, New York, 1963), pp. 213–236.

Herman, B. M.

B. M. Herman, L. J. Battan, “Calculations of Mie backscattering of microwaves from ice spheres,” Quart. J. R. Meteorol. Soc. 87, 223–230 (1961).
[CrossRef]

Kerker, M.

M. Kerker, in The Scattering of Light and Other Electromagnetic Radiation (Academic Press, New York, 1969), p. 68.

Kiehl, J. T.

P. Chylek, J. T. Kiehl, M. K. W. Ko, A. Ashkin, “Surface waves in light scattering by spherical and non-spherical particles,” in Light Scattering by Irregularly Shaped Particles, D. W. Shuerman, ed. (Plenum, New York, 1980), pp. 153–164.
[CrossRef]

Ko, M. K. W.

P. Chylek, J. T. Kiehl, M. K. W. Ko, A. Ashkin, “Surface waves in light scattering by spherical and non-spherical particles,” in Light Scattering by Irregularly Shaped Particles, D. W. Shuerman, ed. (Plenum, New York, 1980), pp. 153–164.
[CrossRef]

Mie, G.

G. Mie, “Beitrage zur Optik truber Median, speziell kolloider Metallosungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, in Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 631.

Nussenzveig, H. M.

Owen, J. F.

Probert-Jones, J. R.

J. R. Probert-Jones, “Surface waves in backscattering and the localization principle,” J. Opt. Soc. Am. 73, 503 (1983).
[CrossRef]

J. R. Probert-Jones, “Surface waves associated with the backscattering of microwave radiation by large ice spheres,” in Interdisciplinary Conference on Electromagnetic Scattering, M. Kerker, ed. (Pergamon, New York, 1963), pp. 237–250.

van de Hulst, H. C.

H. C. van de Hulst, in Light Scattering by Small Particles (Dover, New York, 1981), pp. 373–374.

H. C. van de Hulst, in Light Scattering by Small Particles (Dover, New York, 1981), p. 135.

H. C. van de Hulst, in Light Scattering by Small Particles (Dover, New York, 1981), pp. 155–158.

Ann. Phys. (1)

G. Mie, “Beitrage zur Optik truber Median, speziell kolloider Metallosungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

J. Opt. Soc. Am. (4)

Opt. Lett. (1)

Quart. J. R. Meteorol. Soc. (1)

B. M. Herman, L. J. Battan, “Calculations of Mie backscattering of microwaves from ice spheres,” Quart. J. R. Meteorol. Soc. 87, 223–230 (1961).
[CrossRef]

Other (8)

P. Chylek, J. T. Kiehl, M. K. W. Ko, A. Ashkin, “Surface waves in light scattering by spherical and non-spherical particles,” in Light Scattering by Irregularly Shaped Particles, D. W. Shuerman, ed. (Plenum, New York, 1980), pp. 153–164.
[CrossRef]

H. C. van de Hulst, in Light Scattering by Small Particles (Dover, New York, 1981), p. 135.

M. Kerker, in The Scattering of Light and Other Electromagnetic Radiation (Academic Press, New York, 1969), p. 68.

P. M. Morse, H. Feshbach, in Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 631.

H. C. van de Hulst, in Light Scattering by Small Particles (Dover, New York, 1981), pp. 155–158.

D. Atlas, K. M. Glover, “Back-scatter by dielectric spheres with and without metal caps,” in Interdisciplinary Conference on Electromagnetic Scattering, M. Kerker, ed. (Pergamon, New York, 1963), pp. 213–236.

H. C. van de Hulst, in Light Scattering by Small Particles (Dover, New York, 1981), pp. 373–374.

J. R. Probert-Jones, “Surface waves associated with the backscattering of microwave radiation by large ice spheres,” in Interdisciplinary Conference on Electromagnetic Scattering, M. Kerker, ed. (Pergamon, New York, 1963), pp. 237–250.

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 (6)

Fig. 1
Fig. 1

The real and imaginary parts of the complex-amplitude function for backscattering S(180°) (solid line) and S(180°) −(−1)n(n + ½)bn for n = 34 (dashed line).

Fig. 2
Fig. 2

The real and imaginary parts of (−1)n(n + ½)bn for n = 34.

Fig. 3
Fig. 3

Graphs of |an + bn| for an overlapping pair of spikes for T = ¼, ½ 1, where T = [x(an)−x(bn)]/[w(an) + w(bn)]. The abscissas are in units of [w(an)+ w(bn)] (a) m = 4/3. (b) m = 2.

Fig. 4
Fig. 4

Graphs of T against u/n1/3 for m = 4/3 and m = 2. The graphs are extrapolated below u/n1/3 = 0.4, using the value for u = 0(1), plotted against u/n1/3 = 0.

Fig. 5
Fig. 5

The real and imaginary parts of the complex-amplitude function for forward scattering S(0) and for backward scattering S(180°). The spikes are due to the terms a33 and b34.

Fig. 6
Fig. 6

Graphs of ψn(mkr) (solid line) and ψn′ (mkr) (dashed line) for m = 1.333 and x = 3000.731, which correspond to the an spike with n = 3020, N = 147.

Tables (1)

Tables Icon

Table 1 Values of u0 and u0/n1/3a

Equations (45)

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

I = λ 2 8 π 2 r 2 ( S 1 2 + S 2 2 ) ,
S ( 180° ) = n = 1 ( - 1 ) n ( n + ½ ) ( b n - a n ) .
a n = ψ n ( x ) ψ n ( m x ) - m ψ n ( m x ) ψ n ( x ) ζ n ( x ) ψ n ( m x ) - m ψ n ( m x ) ζ n ( x ) , b n = m ψ n ( x ) ψ n ( m x ) - ψ n ( m x ) ψ n ( x ) n ζ n ( x ) ψ n ( m x ) - ψ n ( m x ) ζ n ( x ) .
ψ n ( x ) = ( π x 2 ) 1 / 2 J n + 1 / 2 ( x ) , χ n ( x ) = - ( π x 2 ) 1 / 2 N n + 1 / 2 ( x ) ,
ζ n ( x ) = ψ n ( x ) + i χ n ( x ) .
p n = ψ n ( x ) ψ n ( m x ) - m ψ n ( m x ) ψ n ( x ) , q n = χ n ( x ) ψ n ( m x ) - m ψ n ( m x ) χ n ( x )
a n = p n p n + i q n = p n 2 p n 2 + q n 2 - i p n q n p n 2 + q n 2 .
tan α n = - p n / q n .
a n spikes : tan ϕ m 2 ρ [ ( 2 u n + ½ ) 2 - 1 4 u ] ,
b n spikes : tan ϕ 1 ρ [ ( 2 u n + ½ ) 2 - 1 4 u ] ,
ϕ ( n + ½ ) ( ρ - tan - 1 ρ ) - ρ u + u 2 2 ( n + ½ ) ρ - π 4 .
a n spikes : tan ϕ m 2 ρ ( 2 u n + ½ ) 1 / 2 , b n spikes : tan ϕ 1 ρ ( 2 u n + ½ ) 1 / 2 ,
p n ( x 0 ) = Δ x d d x q n ( x ) | x = x 0 ,
a n spikes : w 2 m 2 ρ 2 [ ( 2 u 0 n + ½ ) 1 / 2 - 1 4 u 0 ] exp { - [ 4 u 0 3 ( 2 u 0 n + ½ ) 1 / 2 ] } ,
b n spikes : w 2 ρ 2 [ ( 2 u 0 n + ½ ) 1 / 2 - 1 4 u 0 ] exp { - [ 4 u 0 3 ( 2 u 0 n + ½ ) 1 / 2 ] } ,
a n spikes : w 2 m 2 ρ 2 ( 2 u 0 n + ½ ) 1 / 2 exp { - [ 4 u 0 3 ( 2 u 0 n + ½ ) 1 / 2 ] } ,
b n spikes : w 2 ρ 2 ( 2 u 0 n + ½ ) 1 / 2 exp { - [ 4 u 0 3 ( 2 u 0 n + ½ ) 1 / 2 ] } ,
a n spikes :             tan ϕ m 2 ρ ( 1.089 x 1 / 3 + u ) - 1 ,
b n spikes :             tan ϕ 1 ρ ( 1.089 x 1 / 3 + u ) - 1 ,
a n spikes :             w 2 m 2 ρ 2 ( 0.971 x 0 1 / 6 + 0.892 u 0 x 0 - 1 / 6 ) - 2 ,
b n spikes :             w 2 ρ 2 ( 0.971 x 0 1 / 6 + 0.892 u 0 x 0 - 1 / 6 ) - 2 .
x ( a n ) - x ( b n ) = ( 2 u ¯ n + ½ ) 1 / 2 - 1 4 u ¯ + 0 ( n - 1 ) ,             u / n 1 / 3 > 0.6 ,
x ( a n ) - x ( b n ) = ( 2 u ¯ n + ½ ) 1 / 2 + 0 ( n - 1 ) ,             0.2 < u / n 1 / 3 < 0.6 ,
T = x ( a n ) - x ( b n ) w ( a n ) + w ( b n ) .
T ρ 2 2 ( m 2 + 1 ) exp [ 4 2 3 ( u n 1 / 3 ) 3 / 2 ] .
T = 3 4 ( ρ 2 m 2 + 1 ) ,
x ( a n + 1 ) - x ( a n ) x ( b n + 1 ) - x ( b n ) tan - 1 ρ ρ .
w ( a n + 1 ) / w ( a n ) w ( b n + 1 ) / w ( b n ) 1 - [ 2 2 ( u ¯ x ¯ ) 2 - 1 2 u ¯ ] δ u ,
δ u u ( a n + 1 ) - u ( a n ) u ( b n + 1 ) - u ( b n ) 1 - tan - 1 ρ ρ .
tan - 1 ( tan ϕ ) + N π = ϕ .
x ( a n , N ) - x ( a n , N + 1 ) x ( b n , N ) - x ( b n , N + 1 ) π ρ .
Δ x π τ - tan - 1 ρ
Δ n = π / ρ 1 - ( tan - 1 ρ ) / ρ = π ρ - tan - 1 ρ ,
1 + [ 2 2 3 ( u x ¯ ) 3 / 2 - 1 2 x ¯ ] Δ x ,
S ( 0 ) = n = 1 ( n + ½ ) ( a n + b n ) .
E r = H r = 0 , E θ = 1 k r cos ϕ n = 1 i n ( n + ½ ) [ i c n ψ n ( m k r ) - d n ψ n ( m k r ) ] , E ϕ = ( tan ϕ ) E θ , H ϕ = - m k r sin ϕ n = 1 i n ( n + ½ ) [ c n ψ n ( m k r ) - i d n ψ n ( m k r ) ] , H ϕ = - ( cot ϕ ) H θ ,
c n = i / [ ζ n ( x ) ψ n ( m x ) - m ψ n ( m x ) ζ n ( x ) ] , d n = i / [ m ζ n ( x ) ψ n ( m x ) - ψ n ( m x ) ζ n ( x ) ] ,
c n = ( n 2 u ) 1 / 4 ( ρ m 3 ) 1 / 2 exp ( 2 2 3 u 3 / 2 n 1 / 2 )
d n = ( n 2 u ) 1 / 4 ( ρ m ) 1 / 2 exp ( 2 2 3 u 3 / 2 n 1 / 2 )
( n u ) - 1 / 4 exp ( - 2 2 3 u 3 / 2 n 1 / 2 ) .
r a 1 m [ 1 + u n + ( π 8 2 ) 2 / 3 n - 2 / 3 ] ,
r a = 1 m ( 1 + f n - 2 / 3 ) ,
ψ n ( m k r ) = cos [ m k r - ( n + 1 ) ( π / 2 ) ] , ψ n ( m k r ) = - sin [ m k r - ( n + 1 ) ( π / 2 ) ]
E r = i m ( cos ϕ ) ( - i ) n 2 n + 1 n ( n + 1 ) P n 1 ( cos θ ) c n × [ ψ n ( m k r ) + ψ n ( m k r ) ] = 0 , H r = 0 , E θ = i 2 k r cos ϕ 2 n + 1 n ( n + 1 ) τ n c n × [ exp ( - i m k r ) - ( - 1 ) n + 1 exp ( i m k r ) ] , E ϕ = - i 2 k r sin ϕ 2 n + 1 n ( n + 1 ) π n c n × [ exp ( - i m k r ) - ( - 1 ) n + 1 exp ( i m k r ) ] , H θ = i 2 k r m sin ϕ 2 n + 1 n ( n + 1 ) π n c n × [ exp ( - i m k r ) + ( - 1 ) n + 1 exp ( i m k r ) ] , H ϕ = i 2 k r m cos ϕ 2 n + 1 n ( n + 1 ) τ n c n × [ exp ( - i m k r ) + ( - 1 ) n + 1 exp ( i m k r ) ] ,
π n = 1 sin θ P n 1 ( cos θ ) , τ n = d d θ P n 1 ( cos θ ) .

Metrics