JOURNAL OF TIHE OPTICAL SOCIETY OF AMERICA
Transparency of pair-correlated,
VOLUME 65, NUMBER 5
MAY 1975
random distributions of small scatterers, with applications
to the cornea* Victor Twerskyt Mathematics Department, University of Illinois, Chicago, Illinois 60680 (Received 23 October 1974) 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 b > a. The corresponding attenuation coefficients ,1 ccWm 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 13m= io 'kmwith gl0 as the uncorrelated value; WA(W) for spheres decreases more rapidly with increasing w than Wv2 for cylinders, and W2 decreases faster than W,, 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. Index Headings: Scattering; Vision; Transmittance.
We develop simple approximations for the attenuation of light by pair-correlated random distributions of lossless scatterers with small refractive-index contrast, and with average center spacing (2b) very small compared to wavelength (X). In particular, we seek to account for the transparency of biological structures that have globular, cylindrical, or plate-like inclusions (m =3, 2, 1, respectively) in ranges of the parameters where uncorrelated scattering theory suggests that the distributions are opaque, as well as for the opacity that results with swelling. We treat the three m-dimensional scattering problems in parallel, and, in general, use three-dimensional terminology with implicit factors of unit length for m = 2 and unit area for m = 1. The explicit forms that we derive for the dependence of the scattering attenBm- ww m(w) = 8 m on the volume frac-
uation coefficients
tion w (the fraction of space occupied by particles) are quite different than the symmetrical form 3, =w(1- w) = wW, discussed
previously.
1 2
'
Although 8, appeared ap-
is most pronounced. The decrease in W, = 1 - w with increasing w was interpreted for Eq. (130) of Ref. 2 as a decrease of hole effects, with the limit W, - 0 for w- 1 corresponding to perfect transparency (13=0)for a uniform medium of particulate material. We interpret W. - Ws = - Am= - I Am as the effect of the increase of regularity that results from the loss of available space per particle (loss of elbow room) as w increases. Because of their inflexibility, the local order in the arrangement of the neighbors of a given particle must increase with w and consequently the joint distribution of pairs must be statistically correlated; the influence of shape (dimensionality) on local order increases with increasing
m, i. e., we obtain A3>
A2 > Al
We take into account correlations among pairs, as in scattering of x rays by liquids, 3 and represent f3in terms of an integral w[g] of the two-particle (radial) distribution function g. However, in distinction to nu-
merical evaluation or data-inversion programs, we ap-
propriate for flexible particles that could pack to fill the available space, as far as packing is concerned, the
ply our earlier procedure4 (developed for m = 3) to the range b
Transparency of pair-correlated,
VOLUME 65, NUMBER 5
MAY 1975
random distributions of small scatterers, with applications
to the cornea* Victor Twerskyt Mathematics Department, University of Illinois, Chicago, Illinois 60680 (Received 23 October 1974) 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 b > a. The corresponding attenuation coefficients ,1 ccWm 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 13m= io 'kmwith gl0 as the uncorrelated value; WA(W) for spheres decreases more rapidly with increasing w than Wv2 for cylinders, and W2 decreases faster than W,, 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. Index Headings: Scattering; Vision; Transmittance.
We develop simple approximations for the attenuation of light by pair-correlated random distributions of lossless scatterers with small refractive-index contrast, and with average center spacing (2b) very small compared to wavelength (X). In particular, we seek to account for the transparency of biological structures that have globular, cylindrical, or plate-like inclusions (m =3, 2, 1, respectively) in ranges of the parameters where uncorrelated scattering theory suggests that the distributions are opaque, as well as for the opacity that results with swelling. We treat the three m-dimensional scattering problems in parallel, and, in general, use three-dimensional terminology with implicit factors of unit length for m = 2 and unit area for m = 1. The explicit forms that we derive for the dependence of the scattering attenBm- ww m(w) = 8 m on the volume frac-
uation coefficients
tion w (the fraction of space occupied by particles) are quite different than the symmetrical form 3, =w(1- w) = wW, discussed
previously.
1 2
'
Although 8, appeared ap-
is most pronounced. The decrease in W, = 1 - w with increasing w was interpreted for Eq. (130) of Ref. 2 as a decrease of hole effects, with the limit W, - 0 for w- 1 corresponding to perfect transparency (13=0)for a uniform medium of particulate material. We interpret W. - Ws = - Am= - I Am as the effect of the increase of regularity that results from the loss of available space per particle (loss of elbow room) as w increases. Because of their inflexibility, the local order in the arrangement of the neighbors of a given particle must increase with w and consequently the joint distribution of pairs must be statistically correlated; the influence of shape (dimensionality) on local order increases with increasing
m, i. e., we obtain A3>
A2 > Al
We take into account correlations among pairs, as in scattering of x rays by liquids, 3 and represent f3in terms of an integral w[g] of the two-particle (radial) distribution function g. However, in distinction to nu-
merical evaluation or data-inversion programs, we ap-
propriate for flexible particles that could pack to fill the available space, as far as packing is concerned, the
ply our earlier procedure4 (developed for m = 3) to the range b
