Abstract

The scattering of electromagnetic waves by arbitrarily oriented, infinitely long circular cylinders is solved by following the procedures outlined by van de Hulst. The far-field intensities for two cases of a linearly polarized incident wave are derived. The scattering coefficients involve the Bessel functions of the first kind, the Hankel functions of the second kind, and their first derivatives. Calculations are made for ice cylinders at three wavelengths: 0.7 μ, 3 μ, and 10 μ. The numerical results of intensity coefficients are presented as functions of the observation angle ϕ. A significant cross-polarized component for the scattered field, which vanishes only at normal incidence, is obtained. It is also shown that the numerous interference maxima and minima of the intensity coefficients due to single-particle effects depend on the size parameter x as well as on the oblique incident angle α. Since cylinder-type particles are often observed in ice clouds, the light-scattering calculations performed for a circular cylinder in this paper should be of use in the study of cloud microstructure.

© 1972 Optical Society of America

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References

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  1. D. Deirmendjian, Appl. Opt. 3, 187 (1964).
    [Crossref]
  2. J. V. Dave, Appl. Opt. 8, 155 (1969).
    [Crossref] [PubMed]
  3. J. E. Hansen, J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).
    [Crossref]
  4. K. N. Liou, J. E. Hansen, J. Atmos. Sci. 28, 995 (1971).
    [Crossref]
  5. G. Mie, Ann. Phys. 25, 377 (1908).
    [Crossref]
  6. B. J. Mason, Physics of Clouds (Oxford U.P., London, 1957), p. 481.
  7. Lord Rayleigh, Mag. Philos. 36, 365 (1918).
  8. J. R. Wait, Can. J. Phys. 33, 189 (1955).
    [Crossref]
  9. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 415.
  10. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 452.
  11. W. M. Irvine, J. B. Pollack, Icarus 8, 324 (1968).
    [Crossref]
  12. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 1046.

1971 (1)

K. N. Liou, J. E. Hansen, J. Atmos. Sci. 28, 995 (1971).
[Crossref]

1970 (1)

J. E. Hansen, J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).
[Crossref]

1969 (1)

1968 (1)

W. M. Irvine, J. B. Pollack, Icarus 8, 324 (1968).
[Crossref]

1964 (1)

1955 (1)

J. R. Wait, Can. J. Phys. 33, 189 (1955).
[Crossref]

1918 (1)

Lord Rayleigh, Mag. Philos. 36, 365 (1918).

1908 (1)

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

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 1046.

Dave, J. V.

Deirmendjian, D.

Hansen, J. E.

K. N. Liou, J. E. Hansen, J. Atmos. Sci. 28, 995 (1971).
[Crossref]

J. E. Hansen, J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).
[Crossref]

Irvine, W. M.

W. M. Irvine, J. B. Pollack, Icarus 8, 324 (1968).
[Crossref]

Liou, K. N.

K. N. Liou, J. E. Hansen, J. Atmos. Sci. 28, 995 (1971).
[Crossref]

Mason, B. J.

B. J. Mason, Physics of Clouds (Oxford U.P., London, 1957), p. 481.

Mie, G.

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

Pollack, J. B.

J. E. Hansen, J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).
[Crossref]

W. M. Irvine, J. B. Pollack, Icarus 8, 324 (1968).
[Crossref]

Rayleigh, Lord

Lord Rayleigh, Mag. Philos. 36, 365 (1918).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 1046.

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. 452.

Wait, J. R.

J. R. Wait, Can. J. Phys. 33, 189 (1955).
[Crossref]

Ann. Phys. (1)

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

Appl. Opt. (2)

Can. J. Phys. (1)

J. R. Wait, Can. J. Phys. 33, 189 (1955).
[Crossref]

Icarus (1)

W. M. Irvine, J. B. Pollack, Icarus 8, 324 (1968).
[Crossref]

J. Atmos. Sci. (2)

J. E. Hansen, J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).
[Crossref]

K. N. Liou, J. E. Hansen, J. Atmos. Sci. 28, 995 (1971).
[Crossref]

Mag. Philos (1)

Lord Rayleigh, Mag. Philos. 36, 365 (1918).

Other (4)

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 1046.

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

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 452.

B. J. Mason, Physics of Clouds (Oxford U.P., London, 1957), p. 481.

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Figures (6)

Fig. 1
Fig. 1

Geometry for light scattered by an infinitely long cylinder. All symbols are explained in the text.

Fig. 2
Fig. 2

Intensity coefficients i11 (left-hand side) and i22 (right-hand side) as functions of observation angle ϕ for light scattered by a cylinder of circular radius a of 5 μ with incident wavelength λ of 0.7 μ. The upper parts are for oblique incident angles α of 5° (solid line) and 85° (dotted line). The lower parts are for oblique incident angles of 45° (solid line) and 70° (dotted line).

Fig. 3
Fig. 3

Same as Fig. 2, but for cross-polarized intensity coefficient i12.

Fig. 4
Fig. 4

Intensity coefficients i11 (left-hand side) and i22 (right-hand side) as functions of observation angle ϕ for light scattered by a cylinder of circular radius a of 10 μ with incident wavelength λ of 10 μ. Solid lines, dotted lines, and combinations of solid lines and dotted lines are for oblique incident angles α of 5°, 45°, and 85°, respectively.

Fig. 5
Fig. 5

Same as Fig. 4, but the incident wavelength λ is 3 μ.

Fig. 6
Fig. 6

The cross-polarized intensity coefficient i12 as a function of observation angle ϕ for light scattered by a cylinder of circular radius a of 10 μ with incident wavelengths λ of 10 μ (left-hand side) and 3 μ (right-hand side).

Equations (27)

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cos θ = sin 2 α + cos 2 α cos ϕ .
× H = i k m 2 E , × E = i k H , }
2 A + k 2 m 2 A = 0.
2 Ψ + k 2 m 2 Ψ = 0 ,
M Ψ = × ( a z Ψ ) = a r 1 r Ψ ϕ a ϕ Ψ r , m k N Ψ = × M Ψ = a r 2 Ψ z r + a ϕ 1 r 2 Ψ z ϕ a z [ 1 r r ( r Ψ r ) + 1 r 2 2 Ψ ϕ 2 ] , }
E = M υ + i N u , H = m ( M u + i N υ ) . }
1 r r ( r Ψ r ) + 1 r 2 2 Ψ ϕ 2 + 2 Ψ z 2 + m 2 k 2 Ψ = 0.
Ψ n = exp ( i ω t ) Z n ( j r ) exp ( i n ϕ ) exp ( i h z ) ,
E = a r ( i n r υ + h m k u r ) + a ϕ ( υ r i n h m k r u ) + a z [ i ( m 2 k 2 h 2 ) m k u ] , H = a r ( i n m r u + h k υ r ) + a ϕ ( m u r i n h k r υ ) + a z [ i ( m 2 k 2 h 2 ) k υ ] . }
Ψ i = exp [ i ω t i k ( x cos α + z sin α ) ] = n F n J n ( l r ) ,
F n = ( i ) n exp [ i ( ω t + n ϕ h z ) ] , h = k sin α , l = k cos α .
r > a , u i = n F n J n ( l r ) , u s = n b n 1 F n H n ( 2 ) ( l r ) , υ s = n a n 1 F n H n ( 2 ) ( l r ) , r < a , u t = n d n 1 F n J n ( j r ) , υ t = n c n 1 F n J n ( j r ) , }
r > a , υ i = n F n J n ( l r ) , υ s = n a n 2 F n H n ( 2 ) ( l r ) , u s = n b n 2 F n H n ( 2 ) ( l r ) , r < a , u t = n d n 2 F n J n ( j r ) , υ t = n c n 2 F n J n ( j r ) . }
E ϕ i + E ϕ s = E ϕ t , H ϕ i + H ϕ s = H ϕ t , E z i + E z s = E z t , H z i + H z s = H z t , } at r = a .
b n 1 = P n Q n 2 + A n ( ε 1 ) B n ( ε 2 ) Q n 2 + A n ( ε 1 ) A n ( ε 2 ) , a n 2 = P n Q n 2 + B n ( ε 1 ) A n ( ε 2 ) Q n 2 + A n ( ε 1 ) A n ( ε 2 ) , a n 1 = b n 2 = P n Q n A n ( ε 1 ) B n ( ε 1 ) Q n 2 + A n ( ε 1 ) A n ( ε 2 ) , }
A n ( ε 1,2 ) = j H n ( 2 ) ( l a ) H n ( 2 ) ( l a ) ε 1,2 l J n ( j a ) J n ( j a ) , B n ( ε 1,2 ) = j J n ( l a ) J n ( l a ) ε 1,2 l J n ( j a ) J n ( j a ) , { ε 1 = 1 , ε 2 = m 2 , P n = J n ( l a ) / H n ( 2 ) ( l a ) , Q n = i n h ( l 2 j 2 ) / x l j , x = k a = 2 π a / λ , λ = wavelength .
H n ( 2 ) ( l r ) ( 2 / π l r ) 1 2 exp [ i l r + i ( 2 n + 1 ) π / 4 ] , ( l r ) .
u s = ( 2 π l r ) 1 2 exp [ i ( ω t h z l r ) i 3 π / 4 ] n b n 1 cos n ϕ , υ s = ( 2 π l r ) 1 2 exp [ i ( ω t h z l r ) i 3 π / 4 ] n a n 1 sin n ϕ . }
E r s i k sin α cos α u s , E ϕ s i k cos α υ s , E z s i k cos 2 α u s . }
E R = E r cos α + E z sin α , E θ = E r sin α + E z cos α , }
E R s 0 , E θ s i k cos α ( 2 π k R ) 1 2 × exp ( i ω t i k R i 3 π / 4 ) n b n 1 cos n ϕ , E θ s i k cos α ( 2 π k R ) 1 2 × exp ( i ω t i k R i 3 π / 4 ) n a n 1 sin n ϕ , }
I 11 = 2 i 11 I 0 / π k R , I 12 = 2 i 12 I 0 / π k R , } case 1 ,
I 22 = 2 i 22 I 0 / π k R , I 21 = 2 i 21 I 0 / π k R , } case 2 ,
i 11 = | b 01 + 2 n = 1 b n 1 cos n ϕ | 2 , i 12 = | 2 n = 1 a n 1 sin n ϕ | 2 , }
i 22 = | a 02 + 2 n = 1 a n 2 cos n ϕ | 2 , i 21 = | 2 n = 1 b n 2 sin n ϕ | 2 , }
i 12 = i 21 .
H n ( 2 ) ( x ) = J n ( x ) i Y n ( x ) ,

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