Abstract

A novel approach to the estimation of particle size distributions is proposed and investigated. The proposed technique uses only forward scattering data as opposed to known methods using angle scattering. Our approach relies on the fact that, as coherent light travels through the random scattering medium, its coherence deteriorates. The degree of coherence degradation may be described in terms of the mutual intensity function (MIF). The shape of the MIF depends on the distribution of the particle size in the scattering medium. If a coherent receiver is used to detect forward scattered radiation, its output signal depends on the receiver diameter and on the MIF. Dependence of the receiver output signal on the receiver diameter carries information about the MIF and therefore about the particle size distribution. This information may be extracted by means of the techniques suggested in this paper. The suggested techniques are the outgrowth of our previously proposed method of estimating the particle size for the case of a monodispersive medium. As an example, we consider here the estimation of the Junge distribution. However, the method is applicable to other distributions as well. Since the proposed technique is based on an assumption of the form of the distribution, it will yield inaccurate estimates if the actual shape of the distribution function is different from the assumed. The distribution sensitivity of the technique is assessed by considering an extreme case of a flagrant assumption mistake when a monodispersive medium has been assumed while actually the device deals with a Junge-distributed polydispersive medium. This example yields ~50% relative errors in the produced estimates. Since practical distribution deviations should be less serious, the cited number can be considered as an upper bound for the device errors stemming from the distribution sensitivity. The byproducts of our analysis are the development of the MIF for Junge-distributed scattering media and the analysis of the coherent receiver performance for the case when the received light is scattered by Junge-distributed particles.

© 1984 Optical Society of America

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References

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  12. L. G. Kazovsky, Appl. Opt. 23, 448 (1984).
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  17. L. G. Kazovsky, N. S. Kopeika, “The Turbulent Scattering Channel: Effective Heterodyne Receiver Size for Optical Through Millimeter Wavelengths,” in Proceedings, International Conference on Communications, ICC 82, Philadelphia (IEEE, New York, 1982).
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    [CrossRef]
  19. D. L. Fried, Proc. IEEE 55, 57 (1967).
    [CrossRef]
  20. R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976).
  21. D. E. Setzer, “Computed Transmission Through Rain at Microwave and Visible Frequencies,” Bell Syst. Tech. J. 49, 1873 (Oct.1970).
  22. J. O. Laws, D. A. Parsons, “The Relationship of Raindrop Size to Intensity,” Trans. Am. Geophys. Union 24, 452 (1943).
  23. D. E. Kerr, Propagation of Short Radio Waves (Dover, New York, 1951), p. 671.

1984 (1)

1983 (1)

1982 (3)

1980 (3)

1978 (4)

1977 (2)

1975 (1)

1970 (1)

D. E. Setzer, “Computed Transmission Through Rain at Microwave and Visible Frequencies,” Bell Syst. Tech. J. 49, 1873 (Oct.1970).

1967 (2)

G. V. Rosenberg, Atmos. Oceanic Phys. 3, 545 (1967).

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

1955 (1)

C. E. Junge, “Size Distribution and Aging of Natural Aerosols as Determined from Electrical and Optical Data on the Atmosphere,” J. Meteorol. 12, 1 (1955).
[CrossRef]

1943 (1)

J. O. Laws, D. A. Parsons, “The Relationship of Raindrop Size to Intensity,” Trans. Am. Geophys. Union 24, 452 (1943).

Byer, R. L.

Clifford, S. F.

de Wolf, D. A.

Earnshaw, K. B.

Evans, W. H.

Fried, D. L.

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

Gagliardi, R. M.

R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976).

Hansen, M. Z.

Ishimaru, A.

Junge, C. E.

C. E. Junge, “Size Distribution and Aging of Natural Aerosols as Determined from Electrical and Optical Data on the Atmosphere,” J. Meteorol. 12, 1 (1955).
[CrossRef]

Karp, S.

R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976).

Kazovsky, L. G.

L. G. Kazovsky, Appl. Opt. 23, 448 (1984).
[CrossRef] [PubMed]

L. G. Kazovsky, N. S. Kopeika, Appl. Opt. 22, 706 (1983).
[CrossRef] [PubMed]

L. G. Kazovsky, N. S. Kopeika, “The Turbulent Scattering Channel: Effective Heterodyne Receiver Size for Optical Through Millimeter Wavelengths,” in Proceedings, International Conference on Communications, ICC 82, Philadelphia (IEEE, New York, 1982).

Kerr, D. E.

D. E. Kerr, Propagation of Short Radio Waves (Dover, New York, 1951), p. 671.

Kopeika, N. S.

L. G. Kazovsky, N. S. Kopeika, Appl. Opt. 22, 706 (1983).
[CrossRef] [PubMed]

L. G. Kazovsky, N. S. Kopeika, “The Turbulent Scattering Channel: Effective Heterodyne Receiver Size for Optical Through Millimeter Wavelengths,” in Proceedings, International Conference on Communications, ICC 82, Philadelphia (IEEE, New York, 1982).

Lawrence, R. S.

Laws, J. O.

J. O. Laws, D. A. Parsons, “The Relationship of Raindrop Size to Intensity,” Trans. Am. Geophys. Union 24, 452 (1943).

Lerfald, G.

Lutomirski, R. F.

Miller, J. R.

O’Neill, N. T.

Parsons, D. A.

J. O. Laws, D. A. Parsons, “The Relationship of Raindrop Size to Intensity,” Trans. Am. Geophys. Union 24, 452 (1943).

Raskin, V.

Rosenberg, G. V.

G. V. Rosenberg, Atmos. Oceanic Phys. 3, 545 (1967).

Setzer, D. E.

D. E. Setzer, “Computed Transmission Through Rain at Microwave and Visible Frequencies,” Bell Syst. Tech. J. 49, 1873 (Oct.1970).

Tsay, M. K.

Wang, T.

Wolfe, D. C.

Appl. Opt. (14)

Atmos. Oceanic Phys. (1)

G. V. Rosenberg, Atmos. Oceanic Phys. 3, 545 (1967).

Bell Syst. Tech. J. (1)

D. E. Setzer, “Computed Transmission Through Rain at Microwave and Visible Frequencies,” Bell Syst. Tech. J. 49, 1873 (Oct.1970).

J. Meteorol. (1)

C. E. Junge, “Size Distribution and Aging of Natural Aerosols as Determined from Electrical and Optical Data on the Atmosphere,” J. Meteorol. 12, 1 (1955).
[CrossRef]

J. Opt. Soc. Am. (1)

Proc. IEEE (1)

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

Trans. Am. Geophys. Union (1)

J. O. Laws, D. A. Parsons, “The Relationship of Raindrop Size to Intensity,” Trans. Am. Geophys. Union 24, 452 (1943).

Other (3)

D. E. Kerr, Propagation of Short Radio Waves (Dover, New York, 1951), p. 671.

R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976).

L. G. Kazovsky, N. S. Kopeika, “The Turbulent Scattering Channel: Effective Heterodyne Receiver Size for Optical Through Millimeter Wavelengths,” in Proceedings, International Conference on Communications, ICC 82, Philadelphia (IEEE, New York, 1982).

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Figures (5)

Fig. 1
Fig. 1

Diagram of the proposed device.

Fig. 2
Fig. 2

Mutual intensity function M,dB vs ρ/a: the case of a monodispersive medium.

Fig. 3
Fig. 3

Mutual intensity function M,dB vs ρ/a: the case of a Junge-distributed polydispersive medium.

Fig. 4
Fig. 4

Yield Y,dB vs the normalized diaphragm diameter Δ ≙ D/σ, the case of a polydispersive Junge-distributed medium. + denote the points at which condition (43) is satisfied for different values of S; the corresponding values of Δ0 are marked by ↑.

Fig. 5
Fig. 5

Value of Δ0 vs the scattering attenuation S. Small triangles denote numerical solutions of Eq. (49); solid lines correspond to the best least-squares cubic approximation (50).

Tables (1)

Tables Icon

Table I Least-Squares Cubic Approximation of Δ0 vs S Curves

Equations (72)

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Γ ( t 1 , t 2 , r 1 , r 2 ) E [ A ( t 1 , r 1 ) · A * ( t 2 , r 2 ) ] / I S ,
Γ ( t 1 , t 2 , r 1 , r 2 ) = R ( t 1 , t 2 ) · M ( r 1 , r 2 ) ,
M ( ρ ) = { exp ( - ρ 2 · π N Z ) , ρ a , exp ( - a 2 · π N Z ) , ρ a ,
f ( a ) = N · p ( a ) ,
a i δ 2 + ( i - 1 ) · δ ,             i = 1 , 2 , .
n i = ( i - 1 ) δ i δ f ( a ) d a f ( a i ) · δ .
N = n i .
M i ( ρ ) = exp ( - F i · δ ) ,
F i { ρ 2 · π · f ( a i ) · Z , ρ a i , a i 2 · π · f ( a i ) · Z , ρ a i .
M ( ρ ) = i = 1 M i ( ρ ) ,
M ( ρ ) = exp ( - Q ) ,
Q 0 F ( a ) d a ,
F ( a ) { ρ 2 · π · f ( a ) · Z , ρ a , a 2 · π · f ( a ) · Z , ρ a .
Q = 0 ρ F ( a ) d a + ρ F ( a ) d a = π Z · 0 ρ f ( a ) · a 2 · d a + π Z ρ 2 ρ f ( a ) d a .
lim ρ 0 M ( ρ ) = 1.
lim ρ Q = lim ρ π Z 0 ρ f ( a ) a 2 d a = π Z N 0 p ( a ) a 2 d a = π Z N σ 2 ,
σ 2 0 p ( a ) a 2 d a .
lim ρ M ( ρ ) = exp ( - S ) ,
S π Z N σ 2 .
MCF = M ( ρ ) · exp ( - A a Z ) ,
lim ρ 0 MCF = exp ( - A a Z ) ,
lim ρ MCF = exp [ - ( A a + S a ) Z ] ,
S a = 0 σ s ( a ) f ( a ) d a ,
π N σ 2 = 0 σ s ( a ) f ( a ) d a .
σ s ( a ) = π a 2 ,
f ( a ) = { A / a 4 , a min a < , 0 , 0 < a < a min ,
N = 0 f ( a ) d a = a min A a - 4 d a = A / 3 a min 3 ,
σ 2 = 0 p ( a ) a 2 d a = 1 N 0 f ( a ) a 2 d a = 1 N α min A a - 2 d a = A / N a min = 3 a min 2 ,
a min = σ / 3 ,
A = N σ 3 / 3 .
f ( a ) = { N σ 3 / ( a 4 3 ) , σ 3 a < , 0 , 0 < a < σ 3 .
M ( X ) = exp [ - S · V ( X ) ] ,
V ( X ) { X 2 if X < 1 / 3 , 1 - 2 / ( 3 3 X ) if X > 1 / 3 .
Y i ¯ · 2 / ( η A o A ¯ s · π D 2 4 ) ,
Y = 4 2 π D 2 [ 0 D ρ K ( ρ ) M ( ρ ) d ρ ] 1 / 2 ,
K ( ρ ) = { 1 2 [ D 2 arccos ρ D - ρ ( D 2 - ρ 2 ) 1 / 2 ] , p D , 0 , ρ > D .
Y = 4 π [ 0 1 q K n ( q ) M ( q Δ ) d q ] 1 / 2 ,
q ρ D = ρ σ · σ D = X / Δ ,
Δ D / σ ,
K n ( q ) { arccos q - q 1 - q 2 , q 1 , 0 , q 1.
lim Δ 0 Y = 4 π · [ 0 1 q K n ( q ) d q ] 1 / 2 = 1 ,
0 1 ( q arccos q - q 2 1 - q 2 ) d q = π / 16.
lim Δ Y = 4 π · [ 0 1 q K n ( q ) d q ] 1 / 2 · exp ( - S / 2 ) = exp ( - S / 2 ) .
Y min , dB = M min , dB = - S · 10 log e - S · 4.343 ,
Y max , dB = M max , dB = 0 ,
Y min , dB Y ( Δ ) , dB Y max , dB ,
Y max , dB - Y min , dB > 3 dB ,
S > 3 / ( 10 log e ) 0.69.
Y ( Δ 0 ) , dB = Y min , dB + 3 dB .
Δ ˜ 0 = j = 1 4 b j S j - 1 ,
S ^ = 1 4.343 [ Y max , dB - Y min , dB ] = 4.6 log ( Y max / Y min ) ,
S ^ = 4 · 6 · log [ i ¯ min D max 2 / ( i ¯ max D min 2 ) ] ,
Y ˜ , dB 10 log ( C · i ¯ / D 2 )
C = 4 2 / ( π η A O A ¯ S ) ,
Y ˜ ( D = D 0 ) , dB = Y ˜ min , dB + 3 dB .
Y ( Δ 0 ) , dB = Y min , dB + 1 dB .
S > 1 / ( 10 log e ) 0.23.
Δ ^ 0 = j = 1 4 b j S ^ j - 1 ,
σ ^ = D ^ 0 / Δ ^ 0 ,
N ^ = S ^ / π Z σ ^ 2 ,
a ^ min = σ ^ / 3 ,
A ^ = N ^ σ ^ 3 / 3 ,
P r P min · ( D max / D min ) 2 ,
δ σ σ - σ ^ / σ = | D 0 Δ 0 - D ^ 0 Δ ^ 0 | · Δ 0 D 0 = Δ ^ 0 - Δ 0 Δ ^ ,
δ N N - N ^ / N = | S π Z σ 2 - S ^ π Z σ ^ 2 | · π Z σ 2 S = | 1 σ 2 - 1 σ ^ 2 | · σ 2 = | Δ 2 D 0 2 - Δ ^ 2 D ^ 0 2 | · D 0 2 Δ 0 2 = | 1 - ( Δ ^ 0 Δ 0 ) 2 | ,
G ( X , y ) = { X 2 · g ( X , y ) , X < y , y 2 · g ( X , y ) , X > y ,
g ( X , y ) π Z σ 3 · f ( a ) = { π N σ 2 Z / 3 y 4 , 1 3 y < , 0 , 0 < y < 1 / 3 .
Q = 0 G ( X , y ) · 1 σ d a = 0 G ( X , y ) d y = 1 / 3 G ( X , y ) d y ,
Q = π N σ 2 Z 3 · X 2 · 1 / 3 1 y 4 d y = S · X 2 if X < 1 / 3 ,
Q = π N σ 2 Z 3 · ( 1 / 3 X 1 y 2 d y + X X 2 y 4 d y ) = S 3 ( 3 - 2 3 X ) = S [ 1 - 2 3 3 X ] if X > 1 / 3 .
M ( X ) = exp [ - S · V ( X ) ] ,
V ( X ) { X 2 if X < 1 / 3 , 1 - 2 3 3 if X > 1 / 3 .

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