Abstract

Diffracted field contributions to backscattering of an electromagnetic plane wave by a spherical particle are calculated. The diffracted fields give rise to surface waves in the shadow region and can be evaluated by finding surface wave poles and computing their residues. In order to compute the residues the valid range of the Schöbe and Debye asymptotic expansion formulas for the Hankel function is examined. With these asymptotic formulas numerical values of the surface wave complex poles are tabulated. Curves for backscattering cross section due to the first six surface waves are presented as a function of the size parameter ka between 5 and 60 for absorbing spheres of refractive index m = 1.61–i0.0025 as well as nonabsorbing spheres with m = 1.60.

© 1973 Optical Society of America

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References

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  1. G. N. Watson, Proc. Roy. Soc. London A95, 83 (1918).
  2. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 373.
  3. H. C. Bryant, A. J. Cox, J. Opt. Soc. Am. 56, 1529 (1966).
    [CrossRef]
  4. T. S. Fahlen, H. C. Bryant, J. Opt. Soc. Am. 58, 304 (1968).
    [CrossRef]
  5. J. V. Dave, Appl. Opt. 8, 155 (1969).
    [CrossRef] [PubMed]
  6. H. Inada, M. A. Plonus, IEEE Trans. Ant. Prop. AP-18, 649 (1970).
    [CrossRef]
  7. D. Atlas, L. J. Battan, W. G. Harper, B. M. Herman, M. Kerker, E. Matijevic, IEEE Trans. Ant. Prop. AP-11, 68 (1963).
    [CrossRef]
  8. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941). p. 415.
  9. H. Inada, M. A. Plonus, IEEE Trans. Ant. Prop. AP-18, 89 (1970).
    [CrossRef]
  10. R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), p. 93.
  11. G. W. Kattawar, G. N. Plass, Appl. Opt. 6, 1377 (1967).
    [CrossRef] [PubMed]
  12. H. M. Nussenzveig, J. Math. Phys. 10, 82 (1969).
    [CrossRef]
  13. G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge U.P., London, 1966), p. 243.
  14. W. Schöbe, Acta Math. 92, 265 (1954).
    [CrossRef]
  15. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 446.

1970 (2)

H. Inada, M. A. Plonus, IEEE Trans. Ant. Prop. AP-18, 649 (1970).
[CrossRef]

H. Inada, M. A. Plonus, IEEE Trans. Ant. Prop. AP-18, 89 (1970).
[CrossRef]

1969 (2)

H. M. Nussenzveig, J. Math. Phys. 10, 82 (1969).
[CrossRef]

J. V. Dave, Appl. Opt. 8, 155 (1969).
[CrossRef] [PubMed]

1968 (1)

1967 (1)

1966 (1)

1963 (1)

D. Atlas, L. J. Battan, W. G. Harper, B. M. Herman, M. Kerker, E. Matijevic, IEEE Trans. Ant. Prop. AP-11, 68 (1963).
[CrossRef]

1954 (1)

W. Schöbe, Acta Math. 92, 265 (1954).
[CrossRef]

1918 (1)

G. N. Watson, Proc. Roy. Soc. London A95, 83 (1918).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 446.

Atlas, D.

D. Atlas, L. J. Battan, W. G. Harper, B. M. Herman, M. Kerker, E. Matijevic, IEEE Trans. Ant. Prop. AP-11, 68 (1963).
[CrossRef]

Battan, L. J.

D. Atlas, L. J. Battan, W. G. Harper, B. M. Herman, M. Kerker, E. Matijevic, IEEE Trans. Ant. Prop. AP-11, 68 (1963).
[CrossRef]

Bryant, H. C.

Cox, A. J.

Dave, J. V.

Fahlen, T. S.

Harper, W. G.

D. Atlas, L. J. Battan, W. G. Harper, B. M. Herman, M. Kerker, E. Matijevic, IEEE Trans. Ant. Prop. AP-11, 68 (1963).
[CrossRef]

Herman, B. M.

D. Atlas, L. J. Battan, W. G. Harper, B. M. Herman, M. Kerker, E. Matijevic, IEEE Trans. Ant. Prop. AP-11, 68 (1963).
[CrossRef]

Inada, H.

H. Inada, M. A. Plonus, IEEE Trans. Ant. Prop. AP-18, 649 (1970).
[CrossRef]

H. Inada, M. A. Plonus, IEEE Trans. Ant. Prop. AP-18, 89 (1970).
[CrossRef]

Kattawar, G. W.

Kerker, M.

D. Atlas, L. J. Battan, W. G. Harper, B. M. Herman, M. Kerker, E. Matijevic, IEEE Trans. Ant. Prop. AP-11, 68 (1963).
[CrossRef]

Matijevic, E.

D. Atlas, L. J. Battan, W. G. Harper, B. M. Herman, M. Kerker, E. Matijevic, IEEE Trans. Ant. Prop. AP-11, 68 (1963).
[CrossRef]

Newton, R. G.

R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), p. 93.

Nussenzveig, H. M.

H. M. Nussenzveig, J. Math. Phys. 10, 82 (1969).
[CrossRef]

Plass, G. N.

Plonus, M. A.

H. Inada, M. A. Plonus, IEEE Trans. Ant. Prop. AP-18, 89 (1970).
[CrossRef]

H. Inada, M. A. Plonus, IEEE Trans. Ant. Prop. AP-18, 649 (1970).
[CrossRef]

Schöbe, W.

W. Schöbe, Acta Math. 92, 265 (1954).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 446.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941). p. 415.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 373.

Watson, G. N.

G. N. Watson, Proc. Roy. Soc. London A95, 83 (1918).

G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge U.P., London, 1966), p. 243.

Acta Math. (1)

W. Schöbe, Acta Math. 92, 265 (1954).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. Ant. Prop. (3)

H. Inada, M. A. Plonus, IEEE Trans. Ant. Prop. AP-18, 649 (1970).
[CrossRef]

D. Atlas, L. J. Battan, W. G. Harper, B. M. Herman, M. Kerker, E. Matijevic, IEEE Trans. Ant. Prop. AP-11, 68 (1963).
[CrossRef]

H. Inada, M. A. Plonus, IEEE Trans. Ant. Prop. AP-18, 89 (1970).
[CrossRef]

J. Math. Phys. (1)

H. M. Nussenzveig, J. Math. Phys. 10, 82 (1969).
[CrossRef]

J. Opt. Soc. Am. (2)

Proc. Roy. Soc. London (1)

G. N. Watson, Proc. Roy. Soc. London A95, 83 (1918).

Other (5)

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 373.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941). p. 415.

G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge U.P., London, 1966), p. 243.

R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), p. 93.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 446.

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Figures (4)

Fig. 1
Fig. 1

Pole distribution for x = 10 and m = 1.6. The zeros of Eqs. (9) and (10) are shown by dot and cross marks, respectively. Creeping wave poles for a perfectly conducting sphere: the zeros of Hs(2)(x) and [√xHs(2)(x)]′ are plotted by circled dot and cross marks, respectively.

Fig. 2
Fig. 2

Dielectric spheres with m = 1.60 for x between 5 and 60. (a) Backscattering cross sections: Eq. (22) for n = 1,2,3,4,5, and 6. (b) Backscattering cross sections: Eq. (23) for n = 1,2,3,4,5, and 6. (c) Loci of surface wave poles, snA (solid lines) and snB (dashed lines), for n = 1,2,3,4,5, and 6.

Fig. 3
Fig. 3

Dielectric spheres with m = 1.61–i0.0025 for x between 5 and 60. (a) Backscattering cross sections: Eq. (22) for n = 1,2,3,4,5, and 6. (b) Backscattering cross sections: Eq. (23) for n = 1,2,3,4,5, and 6. (c) Loci of surface wave poles, snA (solid lines) and snB (dashed lines), for n = 1,2,3,4,5, and 6.

Fig. 4
Fig. 4

Normalized backscattering cross section for the sum of the first six surface waves for m = 1.61–i0.0025.

Tables (1)

Tables Icon

Table I Values of the Surface Wave Complex Poles

Equations (38)

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E θ = [ exp ( i ω t - i k r ) / i k r ] S ( π ) cos ϕ ,
E ϕ = [ exp ( i ω t - i k r ) / i k r ] S ( π ) sin ϕ ,
S ( π ) = n = 1 ( - 1 ) n ( n + ½ ) ( a n - b n ) .
a n = - { j n ( x ) [ y j n ( y ) ] - j n ( y ) [ x j n ( x ) ] } / { m 2 j n ( y ) [ x h n ( 2 ) ( x ) ] - h n ( 2 ) ( x ) [ y j n ( y ) ] }
b n = - { m 2 j n ( y ) [ x j n ( x ) ] - j n ( x ) [ y j n ( y ) ] } / { m 2 j n ( y ) [ x h n ( 2 ) ( x ) ] - h n ( 2 ) ( x ) [ y j n ( y ) ] } ,
S diff ( π ) = n = 1 [ S n A ( π ) + S n B ( π ) ] ,
S n A ( π ) = i 2 s J s ( y ) x cos ( s π ) H s ( 2 ) ( x ) ( / s ) D M A ( s ) | s = s n A
S n B ( π ) = i 2 m 2 s J s ( y ) x cos ( s π ) H s ( 2 ) ( x ) ( / s ) D M A ( s ) | s = s n B .
D M A ( s ) = m H s ( 2 ) ( x ) [ y j s ( y ) ] - J s ( y ) [ x H s ( 2 ) ( x ) ]
D M B ( s ) = m 2 J s ( y ) [ x H s ( 2 ) ( x ) ] - m H s ( 2 ) ( x ) [ y j s ( y ) ] .
1 cos ( s π ) = 2 l = 1 ( - 1 ) l - 1 exp [ - i ( 2 l - 1 ) s π ] for Im ( s ) < 0
tanh ( y sinh γ - γ s - i π / 4 ) - 1 / m ,
s n A ( 2 m x / π ) - [ ( 2 n - ( 3 / 2 ) ] + ( 2 m / π m 2 ) - i [ ( 2 / π m ) + ( 2 m x / π ) ]             for n = 1 , 2 , 3 , .
x n A ( π / 4 m ) [ 4 n - 3 - ( 4 m / π m 2 ) ] .
coth ( y sinh γ - γ s - i π / 4 ) - 1 / m
s n B ( 2 m x / π ) - [ 2 n - ( 5 / 2 ) ] - ( 2 m / π m 2 ) - i [ ( 2 / π m ) + ( 2 m x / π ) ]             for n = 2 , 3 , 4 , 5 , .
x n B ( π / 4 m ) [ 4 n - 5 + ( 4 m / π m 2 ) ]             n = 2 , 3 , 4 , .
N ( 2 m x / π ) + 1.
s i + 1 = s i - D M ( s ) ( / s ) [ D M ( s ) ] | s = s i ,
s i + 1 - s i < 10 - 6
s 9 A = { 0.0456 - i 0.4899 at x = 16.15 Sch o ¨ be , 0.0481 - i 0.4893 at x = 16.15 Debye , s 9 B = { 0.0711 - i 0.4875 at x = 15.20 Sch o ¨ be , 0.0739 - i 0.4874 at x = 15.20 Debye .
σ π a 2 = 4 x 2 | n = 1 [ S n A ( π ) + S n B ( π ) ] | 2 ,
( σ / π a 2 ) n A = ( 4 / x 2 ) S n A ( π ) 2
( σ / π a 2 ) n B = ( 4 / x 2 ) S n B ( π ) 2
σ π a 2 = 4 x 2 | n = 1 6 [ S n A ( π ) + S n B ( π ) ] | 2
H s ( 1 ) ( 2 ) ( z ) [ e ± θ / ( - i π z sinh γ / 2 ) 1 / 2 ] [ 1 ± ( A 1 / z sinh γ ) + ( 3 A 2 / z 2 sinh 2 γ ) ] ,
θ = z ( sinh γ - γ cosh γ ) - i π / 4 ,
A 1 = 1 8 - 5 24 coth 2 γ ,
A 2 = 3 128 - 77 576 coth 2 γ + 385 3456 coth 4 γ ,
s = z cosh γ = z cosh ( α + i β ) .
H s ( 1 ) ( 2 ) ( z ) / z H s ( 1 ) ( 2 ) ( z ) [ ± sinh γ + ( 1 / 2 z sinh 2 γ ) ± ( 1 / z ) ( / z ) ( A 1 / sinh γ ) ( A 1 / z 2 sinh γ ) + ( 3 / z 2 ) ( / z ) ( A 2 / sinh 2 γ ) - ( 6 A 2 / z 3 sinh 2 γ ) ]
H s ( 1 ) ( 2 ) ( x ) / s H s ( 1 ) ( 2 ) ( x ) [ γ - ( coth γ / 2 z sinh γ ) ± ( 1 / z ) ( / s ) ( A 1 / sinh γ ) + ( 3 / z 2 ) ( / s ) ( A 2 / sinh 2 γ ) ] .
H s ( 1 ) ( 2 ) ( z ) ( 2 / π s tan β ) 1 / 2 exp [ ± i F ( β ) ] [ 1 ( i cot β / 8 s ) ( 1 + 5 3 cot 2 β ) ] ,
F ( β ) = s ( tan β - β ) - π / 4 and s = z cos β .
J s ( z ) P ( s : z ) A i [ Q ( s : z ) ] ,
H s ( 1 ) ( 2 ) ( z ) 2 P ( s : z ) exp ( i π / 3 ) A i [ Q ( s : z ) exp ( ± i 2 π / 3 ) ] ,
P ( s : z ) = ( 2 / z ) 1 / 3 - ( t / 15 ) ( 2 / z ) + ( 13 t 2 / 1260 ) ( 2 / z ) 5 / 3 - [ ( 109 t 3 / 56700 ) + ( 1 / 900 ) ] ( 2 / z ) 7 / 3 + [ ( 203743 t 4 / 523908000 ) + ( 671 t / 7276500 ) ] ( 2 / z ) 3 ,
Q ( s : z ) = t - ( t 2 / 60 ) ( 2 / z ) 2 / 3 + [ ( 2 t 3 / 1575 ) + ( 1 / 140 ) ] ( 2 / z ) 4 / 3 - [ ( 41 t 4 / 283500 ) + ( 4 t / 1575 ) ] ( 2 / z ) 2 + [ ( 6553 t 5 / 327442500 ) + ( 1049 t 2 / 1819125 ) ] ( 2 / z ) 8 / 3 ,

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