Abstract

A scattering model involving complete polarization information for arbitrarily oriented hexagonal columns and plates is developed on the basis of the ray tracing principle which includes contributions from geometric reflection and refraction and Fraunhofer diffraction. We present a traceable and analytic procedure for computation of the scattered electric field and the associated path length for rays undergoing external reflection, two refractions, and internal reflections. We also derive an analytic expression for the scattering electric field in the limit of Fraunhofer diffraction due to an oblique hexagonal aperture. Moreover the theoretical foundation and procedures are further developed for computation of the scattering phase matrix containing 16 elements for randomly oriented hexagonal crystals. Results of the six independent scattering phase matrix elements for randomly oriented large columns and small plates, having length-to-radius ratios of 300/60 and 8/10 μm, respectively, reveal a number of interesting and pronounced features in various regions of the scattering angle when a visible wavelength is utilized in the ray tracing program. Comparisons of the computed scattering phase function, degree of linear polarization, and depolarization ratio for randomly oriented columns and plates with the experimental scattering data obtained by Sassen and Liou for small plates are carried out. We show that the present theoretical results within the context of the geometric optics are in general agreement with the laboratory data, especially for the depolarization ratio.

© 1982 Optical Society of America

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References

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  1. H. Jacobowitz, J. Quant. Spectrosc. Radiat. Transfer 11, 691 (1971).
    [CrossRef]
  2. K. N. Liou, J. Atmos. Sci. 29, 524 (1972).
    [CrossRef]
  3. P. Wendling, R. Wendling, H. K. Weickmann, Appl. Opt. 18, 2663 (1979).
    [CrossRef] [PubMed]
  4. K. Sassen, K. N. Liou, J. Atmos. Sci. 36, 838 (1979).
    [CrossRef]
  5. R. F. Coleman, K. N. Liou, J. Atmos. Sci. 38, 1260 (1981).
    [CrossRef]
  6. D. Clare, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971), p. 56.
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 41, 386.
  8. K. N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980), p. 146.
  9. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 43.
  10. M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954), p. 195.
  11. S. Asano, M. Sato, Appl. Opt. 19, 962 (1980).
    [CrossRef] [PubMed]
  12. K. Sassen, K. N. Liou, J. Atmos. Sci. 36, 852 (1979).
    [CrossRef]
  13. K. N. Liou, H. Lahore, J. Appl. Meteorol. 13, 257 (1974).
    [CrossRef]
  14. K. Sassen, J. Appl. Meteorol. 16, 425 (1977).
    [CrossRef]

1981

R. F. Coleman, K. N. Liou, J. Atmos. Sci. 38, 1260 (1981).
[CrossRef]

1980

1979

P. Wendling, R. Wendling, H. K. Weickmann, Appl. Opt. 18, 2663 (1979).
[CrossRef] [PubMed]

K. Sassen, K. N. Liou, J. Atmos. Sci. 36, 838 (1979).
[CrossRef]

K. Sassen, K. N. Liou, J. Atmos. Sci. 36, 852 (1979).
[CrossRef]

1977

K. Sassen, J. Appl. Meteorol. 16, 425 (1977).
[CrossRef]

1974

K. N. Liou, H. Lahore, J. Appl. Meteorol. 13, 257 (1974).
[CrossRef]

1972

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

1971

H. Jacobowitz, J. Quant. Spectrosc. Radiat. Transfer 11, 691 (1971).
[CrossRef]

Asano, S.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 41, 386.

Clare, D.

D. Clare, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971), p. 56.

Coleman, R. F.

R. F. Coleman, K. N. Liou, J. Atmos. Sci. 38, 1260 (1981).
[CrossRef]

Grainger, J. F.

D. Clare, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971), p. 56.

Jacobowitz, H.

H. Jacobowitz, J. Quant. Spectrosc. Radiat. Transfer 11, 691 (1971).
[CrossRef]

Lahore, H.

K. N. Liou, H. Lahore, J. Appl. Meteorol. 13, 257 (1974).
[CrossRef]

Liou, K. N.

R. F. Coleman, K. N. Liou, J. Atmos. Sci. 38, 1260 (1981).
[CrossRef]

K. Sassen, K. N. Liou, J. Atmos. Sci. 36, 852 (1979).
[CrossRef]

K. Sassen, K. N. Liou, J. Atmos. Sci. 36, 838 (1979).
[CrossRef]

K. N. Liou, H. Lahore, J. Appl. Meteorol. 13, 257 (1974).
[CrossRef]

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

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980), p. 146.

Minnaert, M.

M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954), p. 195.

Sassen, K.

K. Sassen, K. N. Liou, J. Atmos. Sci. 36, 852 (1979).
[CrossRef]

K. Sassen, K. N. Liou, J. Atmos. Sci. 36, 838 (1979).
[CrossRef]

K. Sassen, J. Appl. Meteorol. 16, 425 (1977).
[CrossRef]

Sato, M.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 43.

Weickmann, H. K.

Wendling, P.

Wendling, R.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 41, 386.

Appl. Opt.

J. Appl. Meteorol.

K. N. Liou, H. Lahore, J. Appl. Meteorol. 13, 257 (1974).
[CrossRef]

K. Sassen, J. Appl. Meteorol. 16, 425 (1977).
[CrossRef]

J. Atmos. Sci.

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

K. Sassen, K. N. Liou, J. Atmos. Sci. 36, 852 (1979).
[CrossRef]

K. Sassen, K. N. Liou, J. Atmos. Sci. 36, 838 (1979).
[CrossRef]

R. F. Coleman, K. N. Liou, J. Atmos. Sci. 38, 1260 (1981).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

H. Jacobowitz, J. Quant. Spectrosc. Radiat. Transfer 11, 691 (1971).
[CrossRef]

Other

D. Clare, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971), p. 56.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 41, 386.

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980), p. 146.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 43.

M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954), p. 195.

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Figures (12)

Fig. 1
Fig. 1

Geometry of the orientation of a hexagon with respect to the incident electric vector of a geometric ray. The incident electric vector is defined in the OXYZ′ coordinate, whereas orientation of the hexagon is fixed in the OXYZ coordinate. Points Bi (i = 1, 8) denote the position of the eight vertices of the hexagon corresponding to the aperture cross section for diffraction calculations (also see Fig. 4).

Fig. 2
Fig. 2

Geometry defining the incident, reflected, and refracted rays and angles. Figure 2(a) is for external reflection and first refraction (n = 1). The incident, reflected, and refracted ray paths are defined on the plane BOACO″, and n1 is the normal vector to one of the hexagonal surfaces. Figure 2(b) is for two refractions and internal reflections (n ≥ 2). The incident ray is now in the hexagon. The incident, internally reflected, and refracted ray paths are defined on the plane QTRPS. All the angles and coordinate systems are described in the text.

Fig. 3
Fig. 3

Geometry of the phase shift of the rays undergoing external reflection a1, two refractions a2, and internal reflection an. P0Q0, P1Q1, and PnQn denote planes normal to the direction of these rays

Fig. 4
Fig. 4

Geometry for Fraunhofer diffraction at an arbitrary point p. B i ( i = 1 - 8 ) are the projections of the eight vertices of a hexagonal crystal on the plane normal to an oblique incident ray. θp and ϕp are the polar and azimuthal angles of the diffracted light beam with respect to the OXYZ′ system.

Fig. 5
Fig. 5

Geometry of the scattering by an arbitrarily oriented hexagon in space. The scattering plane is described by ZOP. The incident ray plane is defined by ZOX′. θ and ϕ are the scattering and azimuthal angles for the scattered rays at an arbitrary point P. cosα31, cosα33, and cosα13 are the direction cosines between the axes OZ′ and OX, OZ′ and OZ, and OX′ and OZ, respectively. η(=α33) and ψ2 are orientation angles of the long axis of the crystal (Z axes) with respect to the zenith (OZ′) and azimuthal (OX′) directions. ψ1 is also an orientation angle which is an angle between the normal to the crystal surface (OX) and OQ, where Q is the intersection of the arc CAOQ on the sphere with the normal plane (XY). ψ1 varies from 0 to 2π, but because of the hexagonal symmetry it changes only from 0 to π/3.

Fig. 6
Fig. 6

Contributions of the scattering phase function as a function of the scattering angle for external reflection, two refractions, and internal reflections up to four. The columns with a length-to-radius ratio of 300/60 μm are assumed to be randomly oriented in a horizontal plane when the incident beam with a 0.55-μm wavelength is normal to this plane.

Fig. 7
Fig. 7

Averaged scattering phase function as a function of the scattering angle for randomly oriented columns in a horizontal plane when the incident angles are 90 (normal incidence) 50, and 30°.

Fig. 8
Fig. 8

Averaged degree of linear polarization as a function of the scattering angle for randomly oriented columns in a horizontal plane where the incident angles are 90, 50, and 30°.

Fig. 9
Fig. 9

Angular distribution of six independent elements of the scattering phase matrix for 3-D randomly oriented columns with a length-to-radius of 300/60 μm illuminated by a wavelength of 0.55 μm.

Fig. 10
Fig. 10

Angular distribution of six independent elements of the scattering phase matrix for 3-D randomly oriented plates with a length-to-radius ratio of 8/10 μm illuminated by a laser wavelength of 0.6328 μm.

Fig. 11
Fig. 11

Comparisons of the computed and measured scattering phase functions for randomly oriented columns and plates. The plates observed in a number of scattering and cloud physics experiments have a modal dimension of ~5 μm.

Fig. 12
Fig. 12

Comparisons of the computed and measured degree of linear polarization (upper diagram) and depolarization ratio (lower diagram) for randomly oriented columns and plates. The modal dimension of the plates in the scattering experiments is ~5 μm.

Tables (2)

Tables Icon

Table I Definitions of the Direction Cosines

Tables Icon

Table II Direction Cosines Between O X 1 r Y 1 t Z 1 r and O X 1 t Y 1 t Z 1 t and O X 1 i Y 1 i Z 1 i

Equations (70)

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i = 1 3 cos 2 α i j = 1 ,             j = 1 3 cos 2 α i j = 1 ,             i , j = 1 , 2 , 3.
cos ( n π 3 ) x + sin ( n π 3 ) y - 3 2 a = 0 , n = 0 , 1 , 2 , 3 , 4 , 5 , cos [ ( n - 6 ) π ] z - L 2 = 0 , n = 6 , 7.
cos α n = cos ( n π 3 ) cos β n = sin ( n π 3 ) cos γ n = 0 }             n = 0 , 1 , 2 , 3 , 4 , 5 ,
cos α n = 0 cos β n = 0 cos γ n = cos [ ( n - 6 ) π ] } n = 6 , 7.
[ cos ξ 1 cos ζ 1 cos η 1 ] = A [ cos α 1 cos β 1 cos γ 1 ] ,
A = [ cos α 11 cos α 12 cos α 13 cos α 12 cos α 22 cos α 23 cos α 31 cos α 32 cos α 33 ] .
E 1 i = [ E x 1 i E y 1 i ] = [ cos ϕ 1 i sin ϕ 1 i - sin ϕ 1 i cos ϕ 1 i ] [ E x 0 E y 0 ] ,
cos ϕ 1 i = cos ξ 1 sin η 1 ,             sin ϕ 1 i = cos ζ 1 sin η 1 .
E 1 r = R 1 E 1 i ,             E 1 t = T 1 E 1 i ,
E 1 r = [ E x 1 r E y 1 r ] ,             E 1 t = [ E x 1 t E y 1 t ] ,
R 1 = [ R x 1 0 0 R y 1 ] ,             T 1 = [ T x 1 0 0 T y 1 ] .
R x 1 = - m 2 cos τ 1 i - m 1 cos τ 1 t m 2 cos τ 1 i + m 1 cos τ 1 t , R y 1 = - m 1 cos τ 1 i - m 2 cos τ 1 t m 1 cos τ 1 i + m 2 cos τ 1 t , T x 1 = 2 m 1 cos τ 1 t m 1 cos τ 1 i + m 2 cos τ 1 t , T y 1 = 2 m 1 cos τ 1 t m 1 cos τ 1 i + m 2 cos τ 1 t ,
cos τ 1 i = - cos η 1 sin τ 1 t = sin τ 1 i / m } ,
Ξ 1 r , t = Φ 1 D 1 r , t ,
Ξ 1 r , t = [ cos ξ x 1 r , t cos ξ y 1 r , t cos ξ z 1 r , t cos ζ x 1 r , t cos ζ y 1 r , t cos ζ z 1 r , t cos η x 1 r , t cos η y 1 r , t cos η z 1 r , t ] ,
Φ 1 = [ cos ϕ 1 i - sin ϕ 1 i 0 sin ϕ 1 i cos ϕ 1 i 0 0 0 1 ] .
D 1 r = [ cos 2 τ 1 i 0 sin 2 τ 1 i 0 - 1 0 sin 2 τ 1 i 0 - cos 2 τ 1 i ] ,
D 1 t = [ cos ( τ 1 i - τ 1 t ) 0 - sin ( τ 1 i - τ 1 t ) 0 1 0 sin ( τ 1 i - τ 1 t ) 0 cos ( τ 1 i - τ 1 t ) ] .
E n r = R n E n i             E n t = T n E n i ,
E n i = [ E x n i E y n i ] = { P 2 E 1 t ,             n = 2 , P n - 1 E n - 1 r ,             n 3 ,
E n r = [ E x n r E y n r ] ,             E n t = [ E x n t E y n t ] ,             n 2 ,
P n = [ cos ϕ n i sin ϕ n i - sin ϕ n i cos ϕ n i ] ,             n 2 ,
cos ϕ n i = cos χ n sin ω n ,             sin ϕ n i = cos ψ n sin ω n .
[ cos χ n cos ψ n cos ω n ] = Φ 1 D 1 t Φ 2 D 2 r Φ n - 1 D n - 1 r A [ cos α n cos β n cos γ n ] ,             n 2 ,
Φ n = [ cos ϕ n i - sin ϕ n i 0 sin ϕ n i cos ϕ n i 0 0 0 1 ] ,             n 2 ,
D n r = [ cos 2 τ n i 0 - sin 2 τ n i 0 - 1 0 - sin 2 τ n i 0 - cos 2 τ n i ] ,             n 2 ,
cos τ n i = cos ω n ,             sin τ n t = m sin τ n i .
R x n = - cos τ n i / m + j ( m 2 sin 2 τ n i - 1 ) 1 / 2 cos τ n i / m - j ( m 2 sin 2 τ n i - 1 ) 1 / 2 , R y n = - m cos τ n i + j ( m 2 sin 2 τ n i - 1 ) 1 / 2 m cos τ n i - j ( m 2 sin 2 τ n i - 1 ) 1 / 2 ,
Ξ n r , t = Φ 1 D 1 t Φ 2 D 2 r Φ n - 1 D n - 1 r Φ n D n r , t ,
Ξ n r , t = [ cos ξ x n r , t cos ξ y n r , t cos ξ z n r , t cos ζ x n r , t cos ζ y n r , t cos ζ z n r , t cos η x n r , t cos η y n r , t cos η z n r , t ] ,
D n t = [ cos ( τ n t - τ n i ) 0 - sin ( τ n t - τ n i ) 0 1 0 sin ( τ n t - τ n i ) 0 cos ( τ n t - τ n i ) ] ,
E 1 s = ( S 1 N 1 ) E 1 r ,             E n s = ( S n N n ) E n t ,             n 2 ,
E n s = [ E l n s E r n s ] ,             n = 1 , 2 , ,
S n = [ cos θ n cos ϕ n cos θ n sin ϕ n - sin θ n sin ϕ n - cos ϕ n 0 ] ,             n = 1 , 2 , ,
N 1 = [ cos ξ x 1 r cos ξ y 1 r cos ζ x 1 r cos ζ y 1 r cos η x 1 r cos η y 1 r ] ,             N n = [ cos ξ x n t cos ξ y n t cos ζ x n t cos ζ y n t cos η x n t cos η y n t ] ,             n 2 ,
θ 1 = η z 1 r ,             cos ϕ 1 = cos ξ z 1 r sin η z 1 r ,             sin ϕ 1 = cos ζ z 1 r sin η z 1 r ,
θ n = η z n t ,             cos ϕ n = cos ξ z n t sin η z n t ,             sin ϕ n = cos ζ z n t sin η z n t ,             n 2.
d 0 = x 1 cos α 31 + y 1 cos α 32 + z 1 cos α 33 ,
d 1 = x 1 cos α z 1 r + y 1 cos β z 1 r + z 1 cos γ z 1 r ,
[ cos α z 1 r cos β z 1 r cos γ z 1 r ] = A * [ cos ξ z 1 r cos ζ z 1 r cos η z 1 r ] ,
Δ ϕ 1 = - 2 π λ ( d 0 + d 1 ) ,
d n , n - 1 = [ ( x n - x n - 1 ) 2 + ( y n - y n - 1 ) 2 + ( z n - z n - 1 ) 2 ] 1 / 2 ,
d n = x n cos α z n t + y n cos β z n t + z n cos γ z n t ,
[ cos α z n t cos β z n t cos γ z n t ] = A * [ cos ξ z n t cos ζ z n t cos η z n t ] .
Δ ϕ n = - 2 π λ [ d 0 + d n - m ( d 21 + d 32 + + d n , n - 1 ) ] ,
E s ( θ , ϕ ) = q { n δ ( θ n - θ , ϕ - ϕ n ) w n E n s ( θ n , ϕ n ) × exp [ - j k ( d 0 + d n - m l = 1 n d l + 1 , l ) ] } q ,
δ ( θ - θ n , ϕ - ϕ n ) = { 1 , when θ = θ n , ϕ = ϕ n , 0 , otherwise ,
w n 2 = { 1 ,             n = 1 cos τ n t cos τ 1 t cos τ n i cos τ 1 i m r 2 m r 2 + m i 2 ,             n 2.
u p = - j u 0 λ r B exp ( - j k r ) d x d y ,
[ x B i y B i ] = [ cos α 11 cos α 12 cos α 13 cos α 21 cos α 22 cos α 23 ] [ x B i y B i z B i ] .
u p ( θ p , ϕ p ) = - j u 0 λ r B exp [ - j k ( x cos ϕ p + y sin ϕ p ) sin θ p d x d y ] = - j u 0 k 2 λ r i = 1 8 ( g i P i C i - h i P i D i ) ,
C i = q i + p i a i + 1 ,             D i = q i + P i b i , g i = exp ( - j k D i v i + 1 ) - 1 , h i = exp ( - j k P i u i ) [ exp ( - j k C i v i + 1 ) - 1 ] , P i = sin θ p cos ( ϕ p - ψ i ) ,             q i = sin θ p sin ( ϕ p - ψ i ) , tan ψ i = y B i / x B i , u i = x B i cos ψ i + y B i sin ψ i , v i = - x B i sin ψ i + y B i cos ψ i , a i = u i / v i ,             b i = ( u i + 1 - u i ) / v i + 1 ,             i = 1 , 2 , , 7.
u p ( 0 , 0 ) = - j u 0 λ r B d x d y = - j u 0 2 λ r [ ( x B 8 y B 1 - x B 1 y B 8 ) + i = 1 7 ( x B i y B i + 1 - x B i + 1 y B i ) ] .
E f = [ E x f E x f ] = u p ( θ p , ϕ p ) [ E x 0 E y 0 ] .
E f = u p S f [ E x 0 E y 0 ] ,
S f = [ cos θ p cos ϕ p cos θ p sin ϕ p sin ϕ p - cos ϕ p ] ,
E f = [ E l f E r f ] .
[ E l E r ] Z O P = [ A 2 A 3 A 4 A 1 ] [ E x 0 E y 0 ] Z O X ,
A = A f + A s = [ A 2 f A 2 f A 4 f A 1 f ] + [ A 2 s A 3 s A 4 s A 1 s ] ,
A f = u p r s f ,
A s = q { n δ ( θ - θ n ; ϕ - ϕ n ) w n C n s ( θ n , ϕ n ) × exp [ s - j k ( d 0 + d n - m l = 1 n d l + 1 , l ) ] } q ,
C 1 s = ( S 1 N 1 ) R 1 P 1 , C 2 s = ( S 2 N 2 ) T 2 P 2 T 1 P 1 , C n s = ( S n N n ) T n P n R n - 1 P n - 1 R n - 2 P n - 2 R 2 P 2 T 1 P 1 . }
[ I Q U V ] = F ( θ , ϕ ) [ I 0 Q 0 U 0 V 0 ] ,
F = [ ½ ( M 2 + M 3 + M 4 + M 1 ) ½ ( M 2 - M 3 + M 4 - M 1 ) S 23 + S 41 D 23 + D 41 ½ ( M 2 + M 3 - M 4 - M 1 ) ½ ( M 2 - M 3 - M 4 + M 1 ) S 23 - S 41 D 23 - D 41 S 24 + S 31 S 24 - S 31 S 21 + S 34 D 21 - D 34 D 24 + D 13 D 42 - D 13 D 12 + D 43 S 21 - S 34 ] , M k = A k A k * = A k 2 , S k l = S l k = ½ ( A l A k * + A k A l * ) , - D k l = D l k = j 2 ( A l A k * - A k A l * ) ,             l , k = 1 , 2 , 3 , 4.
P = C F ,
σ s = 0 2 π 0 π ( E l E l * + E r E r * ) sin θ d θ d ϕ .
4 π P 11 ( Ω ) d Ω / 4 π = 1.
P ( θ , ϕ ; η , ψ 2 ) = 1 2 π 0 2 π P ( θ , ϕ ; η , ψ 2 , ψ 1 ) d ψ 1 .
P ( θ ) = 1 4 π 0 2 π 0 π P ( θ , 0 ; η , ψ 2 ) sin η d η d ψ 2 .
P ( θ ) = [ P 11 P 12 0 0 P 12 P 22 0 0 0 0 P 33 - P 43 0 0 P 43 P 44 ] .

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