Abstract

Theory and prototype (at wavelength λ = 5 mm) partial coherence scattering data for optical applications in diagnostic measurements on two component suspensions or gases are presented. Results are given for equal volume mixtures of two sizes of moving randomly distributed large spheres for all realizable values of the fractional volume w (the fraction of the slab region container filled by scattering material). The relative index of refraction of the spheres was about 1.017, and their diameters were 6.52λ and 3.33λ (so that we used about eight times as many small spheres as large ones for each value of w). The spheres were of lightweight Styrofoam, and their motion arose from turbulent air streams (flowing through grids that form the top and bottom of a Styrofoam container), and the data were obtained in real time by processing the instantaneous phase quadrature components of the scattered field with an electronic analog computer. We give results for the forward scattered coherent phase, for the coherent, incoherent, and total intensities, and for the covariant magnitude and phase which (together with the incoherent intensity) provide the variances and covariance of the instantaneous phase quadrature components. We also consider certain reduced data records (from which the major effects of scatterer size and material have been eliminated) to indicate the dependence of the scattering on the fractional volume and to facilitate comparison with earlier data for distributions of identical spheres.

© 1968 Optical Society of America

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References

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  1. V. Twersky, Appl. Opt. 4, 1213 (1965); V. Twersky, in Electromagnetic Scattering, Proceedings of the Interdisciplinary Conference on Electromagnetic Scattering, University of Massachusetts, June 1965, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach Science Publishers, New York, 1967), pp. 579–695. Except for the initial reference in the text, we reserve 1 for citations to the second paper.
    [CrossRef]
  2. C. I. Beard, T. H. Kays, V. Twersky, IEEE Trans. AP-15, 99 (1967).
    [CrossRef]
  3. S. W. Hawley, T. H. Kays, V. Twersky, IEEE Trans. AP-15, 118 (1967).
    [CrossRef]
  4. V. Twersky, in Radio Waves and Circuits, S. Silver, Ed. (Elsevier Publishing Company, Amsterdam, 1963), pp. 233–250; see also Appendix in V. Twersky, J. Opt. Soc. Amer. 52, 145 (1962).
    [CrossRef]
  5. C. I. Beard, T. H. Kays, V. Twersky, Appl. Opt. 4, 1299 (1965).
    [CrossRef]
  6. V. Twersky, IEEE Trans. AP-11, 668 (1963); IEEE Trans. AP-12, 363 (1964).
    [CrossRef]

1967 (2)

C. I. Beard, T. H. Kays, V. Twersky, IEEE Trans. AP-15, 99 (1967).
[CrossRef]

S. W. Hawley, T. H. Kays, V. Twersky, IEEE Trans. AP-15, 118 (1967).
[CrossRef]

1965 (2)

1963 (1)

V. Twersky, IEEE Trans. AP-11, 668 (1963); IEEE Trans. AP-12, 363 (1964).
[CrossRef]

Beard, C. I.

C. I. Beard, T. H. Kays, V. Twersky, IEEE Trans. AP-15, 99 (1967).
[CrossRef]

C. I. Beard, T. H. Kays, V. Twersky, Appl. Opt. 4, 1299 (1965).
[CrossRef]

Hawley, S. W.

S. W. Hawley, T. H. Kays, V. Twersky, IEEE Trans. AP-15, 118 (1967).
[CrossRef]

Kays, T. H.

S. W. Hawley, T. H. Kays, V. Twersky, IEEE Trans. AP-15, 118 (1967).
[CrossRef]

C. I. Beard, T. H. Kays, V. Twersky, IEEE Trans. AP-15, 99 (1967).
[CrossRef]

C. I. Beard, T. H. Kays, V. Twersky, Appl. Opt. 4, 1299 (1965).
[CrossRef]

Twersky, V.

S. W. Hawley, T. H. Kays, V. Twersky, IEEE Trans. AP-15, 118 (1967).
[CrossRef]

C. I. Beard, T. H. Kays, V. Twersky, IEEE Trans. AP-15, 99 (1967).
[CrossRef]

V. Twersky, Appl. Opt. 4, 1213 (1965); V. Twersky, in Electromagnetic Scattering, Proceedings of the Interdisciplinary Conference on Electromagnetic Scattering, University of Massachusetts, June 1965, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach Science Publishers, New York, 1967), pp. 579–695. Except for the initial reference in the text, we reserve 1 for citations to the second paper.
[CrossRef]

C. I. Beard, T. H. Kays, V. Twersky, Appl. Opt. 4, 1299 (1965).
[CrossRef]

V. Twersky, IEEE Trans. AP-11, 668 (1963); IEEE Trans. AP-12, 363 (1964).
[CrossRef]

V. Twersky, in Radio Waves and Circuits, S. Silver, Ed. (Elsevier Publishing Company, Amsterdam, 1963), pp. 233–250; see also Appendix in V. Twersky, J. Opt. Soc. Amer. 52, 145 (1962).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. (3)

V. Twersky, IEEE Trans. AP-11, 668 (1963); IEEE Trans. AP-12, 363 (1964).
[CrossRef]

C. I. Beard, T. H. Kays, V. Twersky, IEEE Trans. AP-15, 99 (1967).
[CrossRef]

S. W. Hawley, T. H. Kays, V. Twersky, IEEE Trans. AP-15, 118 (1967).
[CrossRef]

Other (1)

V. Twersky, in Radio Waves and Circuits, S. Silver, Ed. (Elsevier Publishing Company, Amsterdam, 1963), pp. 233–250; see also Appendix in V. Twersky, J. Opt. Soc. Amer. 52, 145 (1962).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Coherent phase shift α = wk(η − 1)d vs w = 2w1 = 2w2 = w1 + w2 for a binary mixture of spheres; η − 1 = 0.0169, and λ = (2π/k) = 0.52 cm (0.204 in.); the last point w = 0.68 gives an effective container thickness d = 28.3 cm (11.14 in.). The points □ correspond to digital computer averages for static configurations, and the rest correspond to time averages on dynamically varying distributions. The points ○ correspond to the major runs, and the points △ to verification runs.

Fig. 2
Fig. 2

Coherent attenuation coefficient −lnC2 = 2β. The curve u is given by 2βu = 2β0 = 2β0(1 − w) with β0 as in Eq. (10) in terms of the numerical values of the parameters mentioned after Eq. (10). The curve marked h is given by 2βh = 2β0(1 − w) (1 − cw)1/3 = 2β0h with c = 1.374.

Fig. 3
Fig. 3

Coherent intensity C2 corresponding to Fig. 2.

Fig. 4
Fig. 4

Incoherent intensity 〈I2〉. The curve u corresponds to 〈I2〉 = q(1 − C2) with q = 0.475 and C2 of curve u in Fig. 3. The curve marked h is 〈I2〉 = q(1 − C2)h/h with h as for Fig. 2, h = w(1 − w)(1 − cw)2/3 in terms of c = 1.374, and with C2 of curve h in Fig. 3.

Fig. 5
Fig. 5

Total intensity 〈T2〈. The solid points are 〈T2〈 obtained from the T channel, and the open ones are C2 + 〈I2〉 in terms of the data points of Figs. 3 and 4. The curves are C2 + 〈I2〉 from the curves of Figs. 3 and 4.

Fig. 6
Fig. 6

Total attenuation −ln〈T2〉, with data points and curves corresponding to those of Fig. 5.

Fig. 7
Fig. 7

Incoherent intensity divided by the coherent intensity, 〈I2〉/C2, based on values of Figs. 3 and 4.

Fig. 8
Fig. 8

Incoherent intensity divided by unity minus the coherent intensity, 〈I2〉/(1 − C2), based on values of Figs. 3 and 4.

Fig. 9
Fig. 9

Maximum covariance S2. The curve u is given by S2 = p(1 − C2)/2 with C2 of curve u in Fig. 3. Curve h is S2 = p(1 − C2) S h /2h with S h = w(1 − w)(1 − cw) in terms of c = 1.374, and with C2 of curve h in Fig. 3. We use p as in Eq. (20) in terms of the numerical values given after Eq. (21).

Fig. 10
Fig. 10

Maximum values of the correlation coefficient, 2S2/〈I2〉 based on values of Fig. 4 and S2 of Fig. 9.

Fig. 11
Fig. 11

Data for the variant phase s = α + 45° + Φ in degrees. The solid line is α, and the dashed line α + 45° provides the reference for the experimental Φ (equal to approximately 23°).

Fig. 12
Fig. 12

Data for . The curves are based on Eq. (28) with c = 0 for the uniform model and c = 1.374 for the hole model.

Fig. 13
Fig. 13

Data for . The curves are based on Eq. (28) with c = 0 for the uniform model and c = 1.374 for the hole model.

Fig. 14
Fig. 14

Data for S. The curves are based on Eq. (28) with c = 0 for the uniform model and c = 1.374 for the hole model.

Equations (44)

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w = w 1 + w 2 = ( N 1 U 1 + N 2 U 2 ) / V = ρ 1 U 1 + ρ 2 U 2 ,
w 1 = w 2 = w / 2 , N 1 U 1 = N 2 U 2 ,
K - k = 2 π Σ n ρ n f n / k , f n = x ^ · f n ( z ^ , z ^ ) = f n e i Φ n = U n 2 π k 2 ( η n - 1 ) [ 1 + i ( η n - 1 ) k b n ] .
α = ( 2 π d / k ) Σ ρ n Re f n = d k Σ ρ n U n ( η n - 1 ) ,
2 β 0 = ( 4 π d / k ) Σ ρ n Im f n = d 2 k 2 Σ ρ n U n ( η n 2 - 1 ) b n .
α = [ w 1 ( η 1 - 1 ) + w 2 ( η 2 - 1 ) ] k d ,
2 β 0 = [ w 1 ( η 1 - 1 ) 2 b 1 + w 2 ( η 2 - 1 ) 2 b 2 ] 2 k 2 d ,
α = ( w 1 + w 2 ) ( η - 1 ) k d = w ( η - 1 ) k d ,
2 β 0 = ( w 1 b 1 + w 2 b 2 ) ( η - 1 ) 2 2 k 2 d .
2 β 0 = 2 w ( η - 1 ) 2 k 2 d ( b 1 + b 2 ) / 2 = k ( η - 1 ) ( b 1 + b 2 ) α = k ( η - 1 ) ( a 1 + a 2 ) α 3 / 4 ,
2 β 0 = Σ ρ n σ n d = d Σ ρ n π f n ( θ ) 2 d Ω ,
I 2 = q ( γ ) ( 1 - C 2 ) ,
q ( γ ) = γ [ ρ 1 f 1 2 + ρ 2 f 2 2 ] d Ω π [ ρ 1 f 1 2 + ρ 2 f 2 2 ] d Ω = q 1 ρ 1 σ 1 + q 2 ρ 2 σ 2 ρ 1 σ 1 + ρ 2 σ 2 .
q n ( γ ) = γ f n 2 d Ω / σ n ,
ρ n σ n = 2 w n ( η n - 1 ) 2 b n k 2 = w ( η - 1 ) 2 k 2 ( 3 a n / 4 ) ,
q = q 1 a 1 + q 2 a 2 a 1 + a 2 .
2 S 2 e i 2 s p ( 1 - C 2 ) exp [ i ( 2 α + 2 Φ + π / 2 ) ] ,
p e i 2 Φ = π k Z Σ ρ n f n 2 Σ ρ n σ n , 1 Z = 1 d t + 1 d r .
p e i 2 Φ = p 1 ρ 1 σ 1 e i 2 Φ 1 + p 2 ρ 2 σ 2 e i 2 Φ 2 ρ 1 σ 1 + ρ 2 σ 2 = p 1 a 1 e i 2 Φ 1 + p 2 a 2 e i 2 Φ 2 a 1 + a 2 ,
p = [ p 1 2 a 1 2 + p 2 2 a 2 2 + 2 p 1 p 2 a 1 a 2 cos 2 ( Φ 1 - Φ 2 ) ] 1 2 / ( a 1 + a 2 ) ,
tan 2 Φ = p 1 a 1 sin 2 Φ 1 + p 2 a 2 sin 2 Φ 2 p 1 a 1 cos 2 Φ 1 + p 2 a 2 cos 2 Φ 2 .
2 β = 2 β 0 [ 1 - J ( π ) ] , C 2 = e - 2 β ,
I 2 = ( 1 - C 2 ) q ( γ ) [ 1 - J ( γ ) ] [ 1 - J ( π ) ] ,
2 S 2 = ( 1 - C 2 ) p [ 1 - J ( 0 ) ] [ 1 - J ( π ) ] .
J ( γ ) = Σ Σ ρ n ρ m γ f n ( θ ) · f m * ( θ ) [ G n m ( R ) exp [ i k ( r ^ - z ^ ) · R ] d R ] d Ω Σ ρ n γ f n ( θ ) 2 d Ω ,
J ( 0 ) = Σ Σ ρ n ρ m f n f m G n m ( R ) d R Σ ρ n f n 2 w Σ Σ G n m d R 2 ( U 1 + U 2 ) ,
B = w [ 1 - J ( π ) ] ,             I = w [ 1 - J ( γ ) ] ,             S w [ 1 - J ( 0 ) ] ,
B w ( 1 - w ) ( 1 - c w ) 1 3 ,             = w ( 1 - w ) ( 1 - c w ) 2 3 ,             S = w ( 1 - w ) ( 1 - c w ) ,
[ 2 β ] 2 β 0 = [ 2 β ] [ α ] ( 3 / 4 ) k ( η - 1 ) ( a 1 + a 2 ) = B w ,
[ I 2 ] [ 1 - C 2 ] { [ 2 β ] 2 β 0 } 1 q ( γ ) = w ,
[ 2 S 2 ] p [ 1 - C 2 ] { [ 2 β ] 2 β 0 } = S w ,
L = ψ 2 - ψ 2 = Σ Σ u ν u μ - Σ u ν Σ u μ ,
1 n 1 n - 1 n 2 = n 2 - n 2 ,
K [ n 2 - n 2 ] = U 2 ρ [ 1 - ρ G ( R ) d R ] U 2 ρ ( 1 + ρ Γ ) ,
i , j K i j [ 1 n i 1 n j - 1 n i 1 n j ] = K 11 [ n 1 2 - n 1 2 ] + K 22 [ n 2 2 - n 2 2 ] + 2 K 12 [ n 1 n 2 - n 1 n 2 ] .
Σ = ρ 1 U 1 2 ( 1 + ρ 1 Γ 11 ) + ρ 2 U 2 2 ( 1 + ρ 2 Γ 22 ) + 2 ρ 1 ρ 2 U 1 U 1 Γ 12 , Γ i j - G i j ( R ) d R ,
Σ = w 2 ( U 1 + U 2 ) [ 1 + w ( Γ 11 + Γ 22 + 2 Γ 12 ) 2 ( U 1 + U 2 ) ] ;
n i 2 - n i 2 = U i 2 ρ i K i i ( 1 + ρ i Γ i i ) , n 1 n 2 - n 1 n 2 = ρ 1 ρ 2 U 1 U 2 Γ 12 K 12 ,
α = k d ( η - 1 ) w = ( k d / V ) ( η - 1 ) n U ,
α = k d ( η - 1 ) w = k d ( η - 1 ) ( w 1 + w 2 ) = ( k d / V ) ( η - 1 ) ( n 1 U 1 + n 2 U 2 ) ,
α = α e w w e = α e ( w 1 + w 2 ) w e = α e ( N 1 U 1 + N 2 U 2 ) N 1 e U 1 + N 2 e U 2 ,
N 1 U 1 = N 2 U 2 , or w 1 = w 2 = w / 2.
d = N 2 e d 2 + N 1 e d 1 N = ( 1100 ) ( 30 cm ) + ( 8245 ) ( 28.1 cm ) N ,
α = d 1 + d 2 N 2 / N 1 1 + N 2 / N 1 = d 1 + d 2 U 1 / U 2 1 + U 1 / U 2 ,

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