Abstract

We consider transmission through pair-correlated random distributions of lossless dielectric (globular, cylindrical, or plate-like) scatterers with length parameter a and average spacing small compared to wavelength. Each optical particle is centered in a tough adherent transparent coating whose outer surface (sphere, cylinder, or slab) has radius ba. The corresponding attenuation coefficients βWm involve an integral of the appropriate radial-distribution function. Using the scaled-particle equations of state and statistical-mechanics theorems, we evaluate Wm explicitly as a rational function of the volume fraction w of the fluid of rigid b particles. We obtain βm=β0Wm with β0 as the uncorrelated value; W3(w) for spheres decreases more rapidly with increasing w than W2 for cylinders, and W2 decreases faster than W1, the result for slabs. We apply the results for cylinders in terms of W2 to the problem of the transparency of the cornea (whose collagen fibers are the scatterers), as posed by Maurice. The value w ≈ 0.6 gives good accord with the essentials of the data for the transparency of the normal cornea, and the opacity that results from swelling is accounted for by corresponding smaller values of w. Thus, the normal cornea is modeled as a very densely packed two-dimensional gas, with gas-particle (mechanical) radius about 60% greater than the fiber (optical) radius.

© 1975 Optical Society of America

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References

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  1. V. Twersky, J. Opt. Soc. Am. 60, 1084 (1970).
    [CrossRef] [PubMed]
  2. V. Twersky, J. Opt. Soc. Am. 52, 145 (1962).
    [CrossRef] [PubMed]
  3. H. S. Green, The Molecular Theory of Fluids (Interscience, New York, 1952), p. 62ff.
  4. S. W. Hawley, T. H. Kays, and V. Twersky, IEEE Trans. AP-15, 118 (1967).
    [CrossRef]
  5. H. Reiss, H. L. Frisch, and J. L. Lebowitz, J. Chem. Phys. 31, 369 (1959).
    [CrossRef]
  6. E. Helfand, H. L. Frisch, and J. L. Lebowitz, J. Chem. Phys. 34, 1037 (1961).
    [CrossRef]
  7. J. S. Rowlinson, Rep. Progr. Phys. 28, 169 (1965).
    [CrossRef]
  8. G. D. Scott, Nature 187, 908 (1960).
    [CrossRef]
  9. V. Twersky, J. Opt. Soc. Am. 60, 908 (1970).
    [CrossRef]
  10. C. I. Beard, T. H. Kays, and V. Twersky, IEEE Trans. AP-15, 99 (1967).
    [CrossRef]
  11. C. I. Beard, T. H. Kays, and V. Twersky, Appl. Opt. 4, 1299 (1965).
    [CrossRef]
  12. D. M. Maurice, J. Physiol. (Lond.) 136, 263 (1957).
  13. D. M. Maurice, in The Eye, edited by H. Davson (Academic, New York, 1968), Ch. 6.
  14. R. W. Hart and R. A. Farrell, J. Opt. Soc. Am. 59, 766 (1969).
    [CrossRef] [PubMed]
  15. J. L. Cox, R. A. Farrell, R. W. Hart, and M. E. Langham, J. Physiol. 210, 601 (1970).
  16. T. Feuk, IEEE Trans. BME-17, 186 (1970); Scattering of Light in the Corneal Stroma, Tech. Rep. No. 7 (School of Elec. Eng., Chalmers Univ. of Technology, Gotteborg, Sweden, 1970).
  17. G. B. Benedek, Appl. Opt. 10, 459 (1971).
    [CrossRef] [PubMed]
  18. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

1971 (1)

1970 (4)

J. L. Cox, R. A. Farrell, R. W. Hart, and M. E. Langham, J. Physiol. 210, 601 (1970).

T. Feuk, IEEE Trans. BME-17, 186 (1970); Scattering of Light in the Corneal Stroma, Tech. Rep. No. 7 (School of Elec. Eng., Chalmers Univ. of Technology, Gotteborg, Sweden, 1970).

V. Twersky, J. Opt. Soc. Am. 60, 1084 (1970).
[CrossRef] [PubMed]

V. Twersky, J. Opt. Soc. Am. 60, 908 (1970).
[CrossRef]

1969 (1)

1967 (2)

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

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

1965 (2)

1962 (1)

1961 (1)

E. Helfand, H. L. Frisch, and J. L. Lebowitz, J. Chem. Phys. 34, 1037 (1961).
[CrossRef]

1960 (1)

G. D. Scott, Nature 187, 908 (1960).
[CrossRef]

1959 (1)

H. Reiss, H. L. Frisch, and J. L. Lebowitz, J. Chem. Phys. 31, 369 (1959).
[CrossRef]

1957 (1)

D. M. Maurice, J. Physiol. (Lond.) 136, 263 (1957).

Beard, C. I.

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

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

Benedek, G. B.

Cox, J. L.

J. L. Cox, R. A. Farrell, R. W. Hart, and M. E. Langham, J. Physiol. 210, 601 (1970).

Farrell, R. A.

J. L. Cox, R. A. Farrell, R. W. Hart, and M. E. Langham, J. Physiol. 210, 601 (1970).

R. W. Hart and R. A. Farrell, J. Opt. Soc. Am. 59, 766 (1969).
[CrossRef] [PubMed]

Feuk, T.

T. Feuk, IEEE Trans. BME-17, 186 (1970); Scattering of Light in the Corneal Stroma, Tech. Rep. No. 7 (School of Elec. Eng., Chalmers Univ. of Technology, Gotteborg, Sweden, 1970).

Frisch, H. L.

E. Helfand, H. L. Frisch, and J. L. Lebowitz, J. Chem. Phys. 34, 1037 (1961).
[CrossRef]

H. Reiss, H. L. Frisch, and J. L. Lebowitz, J. Chem. Phys. 31, 369 (1959).
[CrossRef]

Green, H. S.

H. S. Green, The Molecular Theory of Fluids (Interscience, New York, 1952), p. 62ff.

Hart, R. W.

J. L. Cox, R. A. Farrell, R. W. Hart, and M. E. Langham, J. Physiol. 210, 601 (1970).

R. W. Hart and R. A. Farrell, J. Opt. Soc. Am. 59, 766 (1969).
[CrossRef] [PubMed]

Hawley, S. W.

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

Helfand, E.

E. Helfand, H. L. Frisch, and J. L. Lebowitz, J. Chem. Phys. 34, 1037 (1961).
[CrossRef]

Kays, T. H.

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

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

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

Langham, M. E.

J. L. Cox, R. A. Farrell, R. W. Hart, and M. E. Langham, J. Physiol. 210, 601 (1970).

Lebowitz, J. L.

E. Helfand, H. L. Frisch, and J. L. Lebowitz, J. Chem. Phys. 34, 1037 (1961).
[CrossRef]

H. Reiss, H. L. Frisch, and J. L. Lebowitz, J. Chem. Phys. 31, 369 (1959).
[CrossRef]

Maurice, D. M.

D. M. Maurice, J. Physiol. (Lond.) 136, 263 (1957).

D. M. Maurice, in The Eye, edited by H. Davson (Academic, New York, 1968), Ch. 6.

Reiss, H.

H. Reiss, H. L. Frisch, and J. L. Lebowitz, J. Chem. Phys. 31, 369 (1959).
[CrossRef]

Rowlinson, J. S.

J. S. Rowlinson, Rep. Progr. Phys. 28, 169 (1965).
[CrossRef]

Scott, G. D.

G. D. Scott, Nature 187, 908 (1960).
[CrossRef]

Twersky, V.

van de Hulst, H. C.

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

Appl. Opt. (2)

IEEE Trans. (3)

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

T. Feuk, IEEE Trans. BME-17, 186 (1970); Scattering of Light in the Corneal Stroma, Tech. Rep. No. 7 (School of Elec. Eng., Chalmers Univ. of Technology, Gotteborg, Sweden, 1970).

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

J. Chem. Phys. (2)

H. Reiss, H. L. Frisch, and J. L. Lebowitz, J. Chem. Phys. 31, 369 (1959).
[CrossRef]

E. Helfand, H. L. Frisch, and J. L. Lebowitz, J. Chem. Phys. 34, 1037 (1961).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Physiol. (1)

J. L. Cox, R. A. Farrell, R. W. Hart, and M. E. Langham, J. Physiol. 210, 601 (1970).

J. Physiol. (Lond.) (1)

D. M. Maurice, J. Physiol. (Lond.) 136, 263 (1957).

Nature (1)

G. D. Scott, Nature 187, 908 (1960).
[CrossRef]

Rep. Progr. Phys. (1)

J. S. Rowlinson, Rep. Progr. Phys. 28, 169 (1965).
[CrossRef]

Other (3)

H. S. Green, The Molecular Theory of Fluids (Interscience, New York, 1952), p. 62ff.

D. M. Maurice, in The Eye, edited by H. Davson (Academic, New York, 1968), Ch. 6.

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

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Equations (52)

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E 0 = x ˆ e i k z , k = 2 π / λ ,
C = e β d ,
β 0 = ρ σ , ρ = N / A d = N / V ,
u f ( r ˆ , z ˆ ) e i k r r ( m 1 ) / 2 , m = 3 , 2 , 1 ,
σ = | f ( r ˆ , z ˆ ) | 2 d Ω ( r ˆ )
β = ρ | F ( r ˆ , z ˆ ) | 2 d Ω , | F ( r ˆ , z ˆ ) | 2 = | f ( r ˆ , z ˆ ) | 2 W ( r ˆ ) , W ( r ˆ ) = 1 + ρ [ g ( R ) 1 ] e i k ( r ˆ z ˆ ) · R d R ,
β = ρ σ W = β 0 W , W = 1 + ρ [ g ( R ) 1 ] d R ,
σ = 3 2 π k 4 υ 3 2 ( η 2 1 η 2 + 2 ) 2 , υ 3 = 4 3 π a 3
σ l = 1 4 k 3 υ 2 2 ( η 1 2 1 ) 2 , σ t = 1 2 k 3 υ 2 2 ( η t 2 1 η t 2 + 1 ) 2 , υ 2 = π a 2 ; σ = 1 2 ( σ l + σ t ) ,
σ = 1 4 k 2 υ 1 2 ( η 2 1 ) 2 , υ 1 = 2 a
T = C + I ( z ˆ ) C + ( 1 C ) q ( α ; z ˆ : W ) = e β d + ( 1 e β d ) q ; q ( α ; z ˆ : W ) = β ( α ) β = ρ β α ; z ˆ | F ( r ˆ , z ˆ ) | 2 d Ω ( r ˆ ) = ρ β α ; z ˆ | f ( r ˆ ; z ˆ ) | 2 W ( r ˆ ) d Ω ,
q ( α ; z ˆ ) = σ ( α ; z ˆ ) / σ = q ( α ; z ˆ ) ,
T C = e β d ,
I ( r ˆ ) ( 1 e β d | sec θ | ) q ( α ; r ˆ : W ) , q ( α ; r ˆ : W ) = ρ α ; r ˆ | F ( r ˆ , z ˆ ) | 2 d Ω ( r ˆ ) / β , | F | 2 = | f | 2 W ( r ˆ ) ,
I 1 ( r ˆ ) = ρ d | sec θ | α ; r ˆ | F ( r ˆ , z ˆ ) | 2 d Ω ( r ˆ ) n ¯ | F ( r ˆ , z ˆ ) | 2 / r m 1 = n ¯ | f ( r ˆ , z ˆ ) | 2 W ( r ˆ ) / r m 1 ,
I ( r ˆ ) = I q ( α ; r ˆ ) , I = 1 e β d | sec θ | q ( α ; r ˆ ) = σ ( α ; r ˆ ) / σ = α ; r ˆ | f ( r ˆ , z ˆ ) | 2 d Ω ( r ˆ ) / σ ,
I ( r ˆ ) = I ( r ˆ i ) = 1 C ( r ˆ ) , C ( r ˆ ) = exp ( β d | sec θ | ) = exp ( β d / | r ˆ z ˆ | ) ,
I ( z ˆ ) = 1 e β d = 1 C , C = C ( z ˆ ) = C ( z ˆ ) ,
I ρ 1 / I ρ 2 = I ρ 1 / I ρ 2 .
I [ λ 1 ] / I [ λ 2 ] = I [ λ 1 ] / I [ λ 2 ] ,
w = ρ υ ; υ 3 = 4 3 π b 3 , υ 2 = π b 2 , υ 1 = 2 b .
ν = ( n n ) 2 = n 2 n 2 = n W ,
ν / n = ρ k B T A ζ T = k B T A ( ρ p ) T = W ,
p / k B T A = ( ρ , υ )
( ρ ) 1 = W ( w )
3 = ρ ( 1 + ρ υ + ρ 2 υ 2 ) ( 1 ρ υ ) 3 , 2 = ρ ( 1 ρ υ ) 2 , 1 = ρ 1 ρ υ .
W 3 = ( 1 w ) 4 ( 1 + 2 w ) 2 , W 2 = ( 1 w ) 3 1 + w , W 1 = ( 1 w ) 2 .
W n 1 2 n w .
W 3 ( 0.63 ) 3.7 × 10 3 , W 2 ( 0.84 ) 2.2 × 10 3 , W 1 ( 1 ) = 0 ,
S m = w W m = w ( 1 w ) m + 1 [ 1 + ( m 1 ) w ] m 1 ; S 3 = w ( 1 w ) 4 ( 1 + 2 w ) 2 , S 2 = w ( 1 w ) 3 1 + w , S 1 = w ( 1 w ) 2 .
w = [ ( 73 ) 1 / 2 7 ] / 12 0.128 , S 3 ( w ) 0.047 ; w = [ ( 7 ) 1 / 2 2 ] / 3 0.215 , S 2 ( w ) 0.086 ; w = 1 3 , S 1 ( w ) 0.148 .
S s S m w 2 ( 1 w ) = W s W m w ( 1 w ) = γ m , γ 3 = 7 + w + w 2 ( 1 + 2 w ) 2 = 1 1 3 W + 1 9 W 2 1 4 3 W + 4 9 W 2 , γ 2 = 3 w 1 + w = 1 + 1 2 W 1 1 2 W , γ 1 = 1 .
β d = ( σ d / υ ) S ( w ) = D 1 S ( w ) , w = w 0 N / N 0 ,
β d = ( ρ σ d ) W ( w ) = D 2 W ( w ) , w = w 0 υ / υ 0 .
β d = ( N σ / A ) W ( w ) = D 3 W ( w ) , w = w 1 d 1 / d = w 1 / t = w t ,
η c 1.47 , η i 1.345 , a 1.55 × 10 5 mm , w c = π ρ a 2 0.23 , d 0.46 mm .
η s w c η c + ν w ν η ν + ( 1 w c ν w ν ) η a = w c η c + ( 1 w c ) η i ,
η i = η a + ν w ν ( η ν η a ) / ( 1 w c ) , η a 1.335 .
η i 1.335 + 8.1 × 10 3 1 w c .
η i 1.335 + 8.1 × 10 3 t 0.23 .
T e β d , β d = δ 0 W 2 , δ 0 = β 0 d = ρ σ 2 d = ρ ( σ l + σ t ) d / 2 .
T = e δ , δ = δ 0 W ( w ) , δ 0 = 2.58 , W = W 2 = ( 1 w ) 3 1 + w , w = w 1 t = w t ,
W 2 ( 0.6 ) 0.040 , T ( 0.6 ) = T 1 e 0.1 0.90 ,
W 2 ( 0.4 ) 0.154 , T 1.5 0.67 .
W 2 ( 0.3 ) 0.264 , T 2 0.51 .
T ( 0.63 ) = T 1 0.92 , T 1.5 0.70 , T 2 0.53 .
T ( 0.57 ) = T 1 0.88 , T 1.5 0.64 , T 2 0.48 ;
T 1 [ 6 ] 0.94 , T 1 [ 5 ] 0.90 , T 1 [ 4 ] 0.82 .
I [ 6 ] 0.06 , I [ 5 ] 0.10 , I [ 4 ] 0.18 .
I [ 6 ] / I [ 4 ] = I [ 6 ] / I [ 4 ] = 1 / 3
I ( r ˆ ) = 1 C ( r ˆ ) , C ( r ˆ ) = exp [ δ 0 ( θ ) W 2 | sec θ | ] , δ 0 ( θ ) = ρ σ 2 ( θ ) d ,
p = k B T A ρ ( 1 ρ υ ) 2 = ( k B T A / υ ) w ( 1 w ) 2 = D t ( t w 1 ) 2 , t = w 1 w .