Abstract

The phase matrix and several quantities for single scattering by an arbitrarily oriented circular cylinder are formulated by using the approximation of ray optics, which includes geometrical reflection and refraction plus Fraunhofer diffraction; then the effects of polarization are considered. Computations were made using electromagnetic wave theory and ray optics approximations for m = 1.31–0.0i and 1.31–0.1i. Results by these methods approach one another as the ratio of the cylinder's circumference to the incident wavelength increases. One of two ray optics approximations proposed requires less computation time than wave theory. The applicability of the ray optics approximation is dependent on the orientation of the cylinder relative to the incident light as well as the size parameter and, moreover, dependent on what quantity for single scattering is compared.

© 1980 Optical Society of America

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References

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  1. H. K. Weickmann, Ber. Dtsch. Wetterdienstes 6, 54 (1949).
  2. A. Heymsfield, J. Atmos. Sci. 32, 799 (1975).
    [Crossref]
  3. Lord Rayleigh, Philos. Mag. 36, 365 (1918).
  4. J. R. Wait, Can. J. Phys. 33, 189 (1955).
    [Crossref]
  5. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  6. M. Kerker, D. D. Cooke, W. A. Farone, R. T. Jacobsen, J. Opt. Soc. Am. 56, 487 (1966).
    [Crossref]
  7. K.-N. Liou, Appl. Opt. 11, 667 (1972).
    [Crossref] [PubMed]
  8. D. Marcuse, H. M. Presby, J. Opt. Soc. Am. 65, 367 (1975).
    [Crossref]
  9. R. D. Birkhoff, J. C. Ashley, H. H. Hubbell, L. C. Emerson, J. Opt. Soc. Am. 67, 564 (1977).
    [Crossref]
  10. T. B. A. Senior, H. Weil, Appl. Opt. 16, 2979 (1977).
    [Crossref] [PubMed]
  11. H. M. Presby, J. Opt. Soc. Am. 64, 280 (1974).
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  12. D. Marcuse, Appl. Opt. 14, 1528 (1975).
    [Crossref] [PubMed]
  13. J. Holoubek, Appl. Opt. 15, 2751 (1976).
    [Crossref] [PubMed]
  14. C. Saekeang, P. L. Chu, J. Opt. Soc. Am. 68, 1298 (1978).
    [Crossref]
  15. A. C. Lind, J. M. Greenberg, J. Appl. Phys. 37, 3195 (1966).
    [Crossref]
  16. K.-N. Liou, J. E. Hansen, J. Atmos. Sci. 28, 995 (1971).
    [Crossref]
  17. K.-N. Liou, J. Atmos. Sci. 29, 524 (1972).
    [Crossref]
  18. D. A. Cross, P. Latimer, J. Opt. Soc. Am. 60, 904 (1970).
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    [Crossref] [PubMed]

1979 (1)

1978 (1)

1977 (2)

1976 (1)

1975 (3)

1974 (1)

1972 (2)

1971 (1)

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

1970 (1)

1966 (2)

1955 (1)

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

1949 (1)

H. K. Weickmann, Ber. Dtsch. Wetterdienstes 6, 54 (1949).

1918 (1)

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

Asano, S.

Ashley, J. C.

Birkhoff, R. D.

Chu, P. L.

Cooke, D. D.

Cross, D. A.

Emerson, L. C.

Farone, W. A.

Greenberg, J. M.

A. C. Lind, J. M. Greenberg, J. Appl. Phys. 37, 3195 (1966).
[Crossref]

Hansen, J. E.

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

Heymsfield, A.

A. Heymsfield, J. Atmos. Sci. 32, 799 (1975).
[Crossref]

Holoubek, J.

Hubbell, H. H.

Jacobsen, R. T.

Kerker, M.

Latimer, P.

Lind, A. C.

A. C. Lind, J. M. Greenberg, J. Appl. Phys. 37, 3195 (1966).
[Crossref]

Liou, K.-N.

K.-N. Liou, J. Atmos. Sci. 29, 524 (1972).
[Crossref]

K.-N. Liou, Appl. Opt. 11, 667 (1972).
[Crossref] [PubMed]

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

Marcuse, D.

Presby, H. M.

Rayleigh, Lord

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

Saekeang, C.

Senior, T. B. A.

van de Hulst, H. C.

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

Wait, J. R.

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

Weickmann, H. K.

H. K. Weickmann, Ber. Dtsch. Wetterdienstes 6, 54 (1949).

Weil, H.

Appl. Opt. (5)

Ber. Dtsch. Wetterdienstes (1)

H. K. Weickmann, Ber. Dtsch. Wetterdienstes 6, 54 (1949).

Can. J. Phys. (1)

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

J. Appl. Phys. (1)

A. C. Lind, J. M. Greenberg, J. Appl. Phys. 37, 3195 (1966).
[Crossref]

J. Atmos. Sci. (3)

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

K.-N. Liou, J. Atmos. Sci. 29, 524 (1972).
[Crossref]

A. Heymsfield, J. Atmos. Sci. 32, 799 (1975).
[Crossref]

J. Opt. Soc. Am. (6)

Philos. Mag. (1)

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

Other (1)

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

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Figures (17)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

(a) Spherical geometry of the externally reflected light. O denotes the incident point. (i), (n0), and (r0) express the incident ray, the normal, and the externally reflected ray, respectively. (b) Spherical geometry of the transmitted light of the first order. (r1), (n1), and (t1) express the refracted ray, the normal, and the transmitted ray of the first order, respectively.

Fig. 3
Fig. 3

Phase functions for a single cylinder at normal incidence. The uppermost, middle, and lowermost curves are those computed from ray optics B, ray optics A, and wave theory, respectively. The vertical scale applies to the lowermost curve. The other curves are successively displaced upward by factors of 10, with the horizontal bars occurring where the phase function has the value of unity.

Fig. 4
Fig. 4

Comparison of ray optics and wave theory for phase functions at normal incidence. The average radii are 5, 10, and 30 μm. The vertical scale applies to the lowermost curves. The other curves are successively displaced upward by factors of 10. Dotted lines (ray optics B) almost overlap dashed lines (ray optics A) and, moreover, overlap solid lines (wave theory) in the forward direction.

Fig. 5
Fig. 5

Same as Fig. 4, but the oblique incident angle α is 45°.

Fig. 6
Fig. 6

Same as Fig. 4, but the oblique incident angle α is 85°.

Fig. 7
Fig. 7

Comparison of ray optics and wave theory for G12. The average radius is 30 μm. The curves for α = 85° are displaced upward by factors of 102, with the horizontal bars occurring where the element has the value 0.1.

Fig. 8
Fig. 8

Comparison of ray optics and wave theory for G11 and G22 at α = 85°. The average radius is 30 μm. The upper curves are displaced upward by factors of 10.

Fig. 9
Fig. 9

Comparison of ray optics and wave theory for the degree of linear polarization at normal incidence, α = 0°. The average radii are 5 and 30 μm. The zero points of each curve are indicated by the horizontal dashed lines.

Fig. 10
Fig. 10

Same as Fig. 9, but the oblique incident angles α are 45 and 85°.

Fig. 11
Fig. 11

Comparison of ray optics and wave theory for G44 at normal incidence, α = 0°. The average radii are 5 and 30 μm. At the observation angle ϕ < 60°, the left vertical scale applies to the lower curves, and the upper curves are displaced upward by factors of 102. At ϕ ≥ 60°, the right vertical scale applies. The zero points of each curve are indicated by the horizontal dashed lines.

Fig. 12
Fig. 12

Comparison of ray optics and wave theory for G43 (= −G34) at normal incidence, α = 0°. The average radii are 5 and 30 μm.

Fig. 13
Fig. 13

Comparison of ray optics and wave theory for the extinction efficiency factor Qext at normal incidence, α = 0°.

Fig. 14
Fig. 14

Same as Fig. 13, but the oblique incident angle α is 85°.

Fig. 15
Fig. 15

Comparison of ray optics and wave theory for phase functions at three oblique incident angles for the case of m = 1.31–0.1i (λ = 0.7 μm). The average radius is 10 μm.

Fig. 16
Fig. 16

Comparison of ray optics and wave theory for the efficiency factors for extinction, scattering, and absorption and the single scattering albedo at α = 0° for the case of m = 1.31–0.1i (λ = 0.7μm).

Fig. 17
Fig. 17

Fractions of energy contained in the transmitted light E(p) (x,m,α) of the different order p for m = 1.31–0.0i (λ = 0.7μm). The numbers in the figure denote p.

Tables (2)

Tables Icon

Table I Extinction Efficiency Factors Averaged over Size Distribution, Eq. (44), for m = 1.31–0.0i (λ = 0.7 μm)

Tables Icon

Table II Efficiency Factors for Extinction, Scattering, and Absorption and Single Scattering Albedo Averaged over Size Distribution for m = 1.31–0.1i (λ = 0.7 μm) and ā = 10 μm

Equations (57)

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( E sl E sr ) = ( 2 π kR cos α ) 1 / 2 exp [ i ( kR + ω t 3 π / 4 ) ] ( T 2 T 3 T 4 T 1 ) ( E il E ir ) ,
A j ( x , m , α , ϕ ) = ( 2 x cos α ) 1 / 2 T j ( x , m , α , ϕ ) , j = 1 , 2 , 3 , 4 ,
A j ( x , m , α , ϕ ) = ( δ 1 j + δ 2 j ) A D + p = 0 N t = 1 M p A j ( p ) [ x , m , α , ϕ ( α , β ) ] ,
A D ( x , α , ϕ ) = ( 2 x cos α ) 1 / 2 sin ( x cos α sin ϕ ) x cos α sin ϕ .
A j ( p ) [ x , m , α , ϕ ( α , β t ) ] = j ( p ) ( π cos β | ϕ / β | ) 1 / 2 exp ( i γ ) , j = 1 , 2 , 3 , 4 ,
( p ) = { L δ 1 R L δ 1 , for p = 0 , L δ 1 T ¯ ( L δ 2 CR ) p 1 L δ 2 T ¯ L δ 1 , for p 1 .
L δ = ( cos δ sin δ sin δ cos δ ) ,
R = ( r 0 0 r ) T ¯ = ( ( 1 r 2 ) 1 / 2 0 0 ( 1 r 2 ) 1 / 2 )
r = m cos τ i cos τ r m cos τ i + cos τ r r = cos τ i m cos τ r cos τ i + m cos τ r } .
cos τ i = cos α cos β ,
m sin τ r = sin τ i .
C = ( 1 0 0 1 ) .
sin δ 1 = sin β / sin τ i ,
cos δ 2 = cos 2 Δ + sin 2 Δ cos ( 2 ξ ) ,
sin Δ = sin α / sin τ i ,
tan ξ = cos Δ tan τ r .
ϕ ( α , β ) = π 2 β p ( π 2 ξ ) .
ϕ ( α , β ) = 2 π k + q ϕ ( α , β ) ,
γ = 3 4 π + δ + [ p + 1 2 ( s 1 ) ] 1 2 π ,
δ = 2 x ( cos τ i p m r cos τ r ) + i 2 π m i λ pl ,
l = 2 a cos τ r 1 sin 2 α / m r 2 ,
s = sgn ( ϕ β ) = ϕ / β | ϕ / β | .
G = ( M 2 1 2 ( M 3 + M 4 ) 0 0 1 2 ( M 3 + M 4 ) M 1 0 0 0 0 S 12 + S 34 D 21 0 0 D 21 S 12 S 34 ) ,
G = ( M 2 M 3 0 0 M 3 M 1 0 0 0 0 S 12 M 3 D 21 0 0 D 21 S 12 + M 3 ) .
G 44 G 33 = 2 G 12 ( 0 )
G = 1 2 ( G 11 + G 12 + G 21 + G 22 ) ,
DP = ( G 11 + G 12 ) ( G 21 + G 22 ) ( G 11 + G 12 ) + ( G 21 + G 22 ) .
Q sca = 1 2 ( Q sca , l + Q sca , r ) = 1 2 [ 1 π 0 π ( G 11 + G 12 ) d ϕ + 1 π 0 π ( G 21 + G 22 ) d ϕ ] = 1 π 0 π Gd ϕ .
C sca = 2 a cos α Q sca .
Q abs ( x , m , α ) = 1 2 ( Q abs , l + Q abs , r ) = 1 2 ( { 1 2 p = 0 N [ E 2 ( p ) + E 4 ( p ) ] } + { 1 2 p = 0 N [ E 3 ( p ) + E 1 ( p ) ] } ) = 1 p = 0 N E ( p ) ,
E ( p ) = j = 1 4 E j ( p ) = j = 1 4 0 π / 2 | j ( p ) | 2 exp ( 4 π m i λ pl ) cos β d β .
Q ext = Q sca + Q abs .
G kl ( m , α , ϕ ) = G kl ( x , m , α , ϕ ) xn ( x ) dx / xn ( x ) dx , k , l = 1 , 2 , 3 , 4 ,
G kl ( x , m , α , ϕ ) = G kl D ( x , α , ϕ ) + p = 0 N t = 1 M p G kl ( p ) [ x , m , α , ϕ ( α , β t ) ] , k , l = 1 , 2 , 3 , 4 ,
Q ext = 2.0 Q sca = 1 2 ( Q sca , l + Q sca , r ) = 1 2 ( { 1 + 2 p = 0 N [ E 2 ( p ) + E 4 ( p ) ] } + { 1 + 2 p = 0 N [ E 3 ( p ) + E 1 ( p ) ] } ) = 1 + p = 0 N E ( p ) } .
A j = s = 0 K a j ( s ) exp { i [ δ j ( s ) + σ ( s ) ] } , j = 1 , 2 , 3 , 4 ,
D 21 = s = 0 K a 1 ( s ) a 2 ( s ) sin [ δ 1 ( s ) δ 2 ( s ) ] + t = 0 K t = 0 K ( t u ) a 1 ( t ) a 2 ( u ) sin [ δ 1 ( t ) δ 2 ( u ) + σ ( t ) σ ( u ) ] .
T 1 ( x , m , α , ϕ ) = a 02 + 2 n = 1 a n 2 cos n ϕ ,
T 2 ( x , m , α , ϕ ) = b 01 + 2 n = 1 b n 1 cos n ϕ ,
T 3 ( x , m , α , ϕ ) = 2 n = 1 b n 2 sin n ϕ ,
T 4 ( x , m , α , ϕ ) = 2 n = 1 a n 1 sin n ϕ .
Q ext = 1 2 ( Q ext , l + Q ext , r ) = 1 2 2 x cos α { [ Re ( b 01 ) + 2 n = 1 Re ( b n 1 ) ] + [ Re ( a 02 ) + 2 n = 1 Re ( a n 2 ) ] }
Q sca = 1 2 ( Q sca , l + Q sca , r ) = 1 2 2 x cos α { [ | b 01 | 2 + 2 n = 1 ( | b n 1 | 2 + | a n 1 | 2 ) ] + [ | a 02 | 2 + 2 n = 1 ( | a n 2 | 2 + | b n 2 | 2 ) ] } .
Q ¯ sca = Q sca xn ( x ) dx / xn ( x ) dx .
n ( x ) x 6 exp ( bx ) ,
( 0 ) r ( cos ϕ sin ϕ sin ϕ cos ϕ ) .
r p 1 ( 1 r ) 2 ,
Q ¯ ext
Q ¯ sca
Q ¯ abc
ω ¯
u s = i R λ u i a a exp { i [ ω t k ( X cos α + Z sin α ) k r ¯ ] } dYdZ .
r ¯ = R ( 1 2 Y R cos α sin ϕ 2 Z R sin α + Y 2 + Z 2 R 2 ) 1 / 2 R Y cos α sin ϕ Z sin α + Z 2 2 R .
E s = ( 2 x cos α ) 1 / 2 sin ( x cos α sin ϕ ) x cos α sin ϕ ( x π kR ) 1 / 2 × exp [ i ( ω t kR 3 4 π ) ] E i .
ω j ( p ) = 1 π 0 π | A j ( p ) | 2 d ϕ , j = 1 , 2 , 3 , 4 .
η j ( p ) exp { i arg [ A j ( p ) ] } , for [ η j ( p ) ] 2 0 0 , for [ η j ( p ) ] 2 < 0 } ,
η j ( p ) = { | A j ( p ) | 2 [ ω j ( p ) E j ( p ) ] / Δ w } 1 / 2 ,

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