Abstract

The present work relates the turbidity of the eye to microscopic spatial fluctuations in itsindex of refraction. Such fluctuations are indicated in electron microscope photographs. Byexamining the superposition of phases of waves scattered from each point in the medium, we provide amathematical demonstration of the Bragg reflection principle which we have recently used in theinterpretation of experimental investigations: namely, that the scattering of light is produced onlyby those fluctuations whose fourier components have a wavelength equal to or larger than one halfthe wavelength of light in the medium. This consideration is applied first to the scattering oflight from collagen fibers in the normal cornea. We demonstrate physically and quantitatively that alimited correlation in the position of near neighbor collagen fibers leads to corneal transparency.Next, the theory is extended to predict the turbidity of swollen, pathologic corneas, wherein thenormal distribution of collagen fibers is disturbed by the presence of numerouslakes—regions where collagen is absent. A quantitative expression for theturbidity of the swollen cornea is given in terms of the size and density of such lakes. Finally,the theory is applied to the case of the cataractous lens. We assume that the cataracts are producedby aggregation of the normal lens proteins into an albuminoid fraction and provide a formula for thelens turbidity in terms of the molecular weight and index of refraction of the individual albuminoidmacromolecules. We provide a crude estimate of the mean albuminoid molecular weight required forlens opacity.

© 1971 Optical Society of America

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References

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  1. D. M. Maurice, J. Physiol. (London) 136, 263 (1957).
  2. W. Schwarz, Z. Zellforsch. 38, 26 (1953).
    [CrossRef]
  3. M. A. Jakus, “The Fine Structure of the HumanCornea,” in The Structure of the Eye, G. K. Smelser, Ed. (Academic, NewYork, 1961).
  4. J. N. Goldman, G. B. Benedek, Invest. Ophthalmol. 6, 574 (1967).
    [PubMed]
  5. J. N. Goldman, G. B. Benedek, C. H. Dohlman, B. Kravitt, Invest. Ophthalmol. 7, 501 (1968).
    [PubMed]
  6. T. Feuk, IEEE Trans. Biomed. Eng. (in press) (1970) andprivate communication.
  7. R. W. Hart, R. A. Farrell, J. Opt. Soc. Amer. 59, 766 (1969).
    [CrossRef]
  8. H. C. Van de Hulst, Light Scattering by Small Particles(WileyNew York, 1957).
  9. R. W. Hart, R. A. Farrell, Appl. Phys. Lab., Johns Hopkins Univ., privatecommunication.
  10. In Fig. 10 we indicate that in the region of the lakes theindex of refraction is the same as that of the ground substance. This is probably not quite correctas there is likely to be water in these lakes. This would tend to lower the index of refraction ofthe lake to a value closer to that of water. The discussion we give above can be very simplyextended to include this effect. We shall neglect this effect as it does not alter substantially theline of argument presented above.
  11. M. Abramowitz, I. Stegun, Eds. Handbook of Mathematical Functions(Dover, New York,1965), p. 364, Eq.9.2.1; p. 370, Eq. 9.4.4.
  12. B. Phillipson, Acta Ophthalmol. (Stockholm) Suppl.103 (1969); see also Acta Ophthalmol. 47, 1089 (1969).
  13. A. Spector, Invest. Ophthalmol. 4, 579 (1965).
  14. S. Trokel, Invest. Ophthalmol. 1, 493 (1962).
  15. D. McIntyre, F. Gornick, Eds., Light Scattering from Dilute Polymer Solutions(Gordon and Breach, New York,1964) (see, for example, article by W. Heller, p. 41).
  16. J. Kinoshita, Howe Laboratory, Harvard Medical School, privatecommunication.
  17. B. Phillipson, Invest. Ophthalmol. 8, 281 (1969) (especially p. 288); see also Invest. Ophthalmol. 8, 271 (1969).
    [PubMed]

1969

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

B. Phillipson, Acta Ophthalmol. (Stockholm) Suppl.103 (1969); see also Acta Ophthalmol. 47, 1089 (1969).

B. Phillipson, Invest. Ophthalmol. 8, 281 (1969) (especially p. 288); see also Invest. Ophthalmol. 8, 271 (1969).
[PubMed]

1968

J. N. Goldman, G. B. Benedek, C. H. Dohlman, B. Kravitt, Invest. Ophthalmol. 7, 501 (1968).
[PubMed]

1967

J. N. Goldman, G. B. Benedek, Invest. Ophthalmol. 6, 574 (1967).
[PubMed]

1965

A. Spector, Invest. Ophthalmol. 4, 579 (1965).

1962

S. Trokel, Invest. Ophthalmol. 1, 493 (1962).

1957

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

1953

W. Schwarz, Z. Zellforsch. 38, 26 (1953).
[CrossRef]

Benedek, G. B.

J. N. Goldman, G. B. Benedek, C. H. Dohlman, B. Kravitt, Invest. Ophthalmol. 7, 501 (1968).
[PubMed]

J. N. Goldman, G. B. Benedek, Invest. Ophthalmol. 6, 574 (1967).
[PubMed]

Dohlman, C. H.

J. N. Goldman, G. B. Benedek, C. H. Dohlman, B. Kravitt, Invest. Ophthalmol. 7, 501 (1968).
[PubMed]

Farrell, R. A.

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

R. W. Hart, R. A. Farrell, Appl. Phys. Lab., Johns Hopkins Univ., privatecommunication.

Feuk, T.

T. Feuk, IEEE Trans. Biomed. Eng. (in press) (1970) andprivate communication.

Goldman, J. N.

J. N. Goldman, G. B. Benedek, C. H. Dohlman, B. Kravitt, Invest. Ophthalmol. 7, 501 (1968).
[PubMed]

J. N. Goldman, G. B. Benedek, Invest. Ophthalmol. 6, 574 (1967).
[PubMed]

Hart, R. W.

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

R. W. Hart, R. A. Farrell, Appl. Phys. Lab., Johns Hopkins Univ., privatecommunication.

Jakus, M. A.

M. A. Jakus, “The Fine Structure of the HumanCornea,” in The Structure of the Eye, G. K. Smelser, Ed. (Academic, NewYork, 1961).

Kinoshita, J.

J. Kinoshita, Howe Laboratory, Harvard Medical School, privatecommunication.

Kravitt, B.

J. N. Goldman, G. B. Benedek, C. H. Dohlman, B. Kravitt, Invest. Ophthalmol. 7, 501 (1968).
[PubMed]

Maurice, D. M.

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

Phillipson, B.

B. Phillipson, Acta Ophthalmol. (Stockholm) Suppl.103 (1969); see also Acta Ophthalmol. 47, 1089 (1969).

B. Phillipson, Invest. Ophthalmol. 8, 281 (1969) (especially p. 288); see also Invest. Ophthalmol. 8, 271 (1969).
[PubMed]

Schwarz, W.

W. Schwarz, Z. Zellforsch. 38, 26 (1953).
[CrossRef]

Spector, A.

A. Spector, Invest. Ophthalmol. 4, 579 (1965).

Trokel, S.

S. Trokel, Invest. Ophthalmol. 1, 493 (1962).

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles(WileyNew York, 1957).

Acta Ophthalmol. (Stockholm) Suppl.

B. Phillipson, Acta Ophthalmol. (Stockholm) Suppl.103 (1969); see also Acta Ophthalmol. 47, 1089 (1969).

Invest. Ophthalmol.

A. Spector, Invest. Ophthalmol. 4, 579 (1965).

S. Trokel, Invest. Ophthalmol. 1, 493 (1962).

B. Phillipson, Invest. Ophthalmol. 8, 281 (1969) (especially p. 288); see also Invest. Ophthalmol. 8, 271 (1969).
[PubMed]

J. N. Goldman, G. B. Benedek, Invest. Ophthalmol. 6, 574 (1967).
[PubMed]

J. N. Goldman, G. B. Benedek, C. H. Dohlman, B. Kravitt, Invest. Ophthalmol. 7, 501 (1968).
[PubMed]

J. Opt. Soc. Amer.

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

J. Physiol. (London)

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

Z. Zellforsch.

W. Schwarz, Z. Zellforsch. 38, 26 (1953).
[CrossRef]

Other

M. A. Jakus, “The Fine Structure of the HumanCornea,” in The Structure of the Eye, G. K. Smelser, Ed. (Academic, NewYork, 1961).

T. Feuk, IEEE Trans. Biomed. Eng. (in press) (1970) andprivate communication.

H. C. Van de Hulst, Light Scattering by Small Particles(WileyNew York, 1957).

R. W. Hart, R. A. Farrell, Appl. Phys. Lab., Johns Hopkins Univ., privatecommunication.

In Fig. 10 we indicate that in the region of the lakes theindex of refraction is the same as that of the ground substance. This is probably not quite correctas there is likely to be water in these lakes. This would tend to lower the index of refraction ofthe lake to a value closer to that of water. The discussion we give above can be very simplyextended to include this effect. We shall neglect this effect as it does not alter substantially theline of argument presented above.

M. Abramowitz, I. Stegun, Eds. Handbook of Mathematical Functions(Dover, New York,1965), p. 364, Eq.9.2.1; p. 370, Eq. 9.4.4.

D. McIntyre, F. Gornick, Eds., Light Scattering from Dilute Polymer Solutions(Gordon and Breach, New York,1964) (see, for example, article by W. Heller, p. 41).

J. Kinoshita, Howe Laboratory, Harvard Medical School, privatecommunication.

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Figures (12)

Fig. 1
Fig. 1

Schematic diagram of arrangement of collagen fibers in a lamella. The incident light is shownpropagating along the x direction.

Fig. 2
Fig. 2

Schematic diagram showing the positions of the observation point (or field point) and the source point Rj forfibers in a lamella.

Fig. 3
Fig. 3

Direction of the polarization vector of the incident light field (E0)and incident propagation direction (x). Also shown are the unit vectors 1 ^ z and 1 ^ θ that are used to specify the polarization of the scattered field asobserved a distance r from the axis of a single scattering collagen.

Fig. 4
Fig. 4

Geometric representation of the difference in optical path between scattering from a fibersituated at the origin O, and one situated at the source pointRj. The scattered direction is specified by the scatteringangle θ.

Fig. 5
Fig. 5

Geometry of the scattering process. The wave vector of the incident light isk0, the wave vector of the scattered light is k. The scatteringvector is the difference vector. Its length is 2k0sinθ/2.

Fig. 6
Fig. 6

The total scattered electric field at some particular field point () isthe sum of the fields radiated from each fiber. The value of the sum depends on the phases of eachof the constituent waves, as is indicated above.

Fig. 7
Fig. 7

Representation of the random spatial fluctuation in the fiber densityΔρ(R) as a function of position(R) inside the cornea.

Fig. 8
Fig. 8

General form for the conditional number densityρ(R″|0). When R″ becomesappreciably larger than the correlation range Rc, theconditional probability becomes equal to the mean density〈ρ〉. For values substantially less than the correlationrange ρ(R″|0) is zero.

Fig. 9
Fig. 9

General form for the function f(R″) = 1−ρ(R″|0)/〈ρ〉.

Fig. 10
Fig. 10

Electron microscope photograph of swollen pathologic corneal stroma from paper in Ref. 5. The arrows point to some of the lakes where the collagen fibersare not present. The short scale marker has the length of 2000 Å.

Fig. 11
Fig. 11

Characterization of the scattering from lakes. In (a) we represent the fluctuation in index ofrefraction as a function of position in a lamella containing lakes. Each line represents a collagenfiber and the gaps represent the lakes. In (b) and (c) we represent an arrangement of scatteringamplitudes which will radiate the same field as would be radiated from the fiber arrangement in (a).In (b) the missing fibers are randomly replaced with the average fiber density in the region of thelakes. In (c) fibers with negative scattering amplitudes cancel the field radiated by the replacedfibers. The field radiated by the sum of the configuration (b) plus (c) is the same as that radiatedby the original swollen cornea represented in (a).

Fig. 12
Fig. 12

A plot of the function[2J1(Kaα)/(Kaα)]2vs Kaα.

Equations (79)

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E j ( R , t ) = E 0 j e i ( k 0 R - ω 0 t ) e i ( k 0 - k ) · R j .
E 0 = E 0 4 ( λ r ) 1 2 ( 2 π r 0 λ ) 2 ( m 2 - 1 ) [ 1 ^ z cos γ + 1 ^ θ ( 2 m 2 + 1 ) sin γ cos θ ] ,
Δ Φ = [ 2 π / ( λ / n ) ] R j ( cos θ - cos φ ) .
k 0 · R j = [ 2 π / ( λ / n ) ] R j cos φ ,
k · R j = [ 2 π / ( λ / n ) ] R j cos θ ,
Δ Φ = ( k - k 0 ) · R j = K · R j .
K = 2 k 0 sin θ / 2 = [ 4 π / ( λ / n ) ] sin ( θ / 2 ) .
E tot ( R , t ) = j = 1 N E j ( R , t ) = j = 1 N E 0 e i K · R j e i k 0 R - ω 0 t ,
E tot ( R , t ) = E 0 e i k 0 R - ω 0 t j = 1 N e i K · R j .
I j = 1 N e i K · R j .
A d 2 R δ ( R - R j )
I = A d 2 R e i K · R ( j = 1 N δ ( R - R j ) ) .
j = 1 N δ ( R - R j ) = ρ ( R ) .
ρ ( R ) = ρ + Δ ρ ( R ) ,
I = A d 2 R e i K · R ρ + d 2 R e i K · R Δ ρ ( R ) .
Δ ρ ( R ) = 1 2 π A d 2 q e iq · R Δ ρ ( q ) .
Δ ρ ( q ) = 1 2 π A d 2 R e i q · R Δ ρ ( R ) .
2 π / λ f = K = [ 4 π / ( λ / n ) ] sin 1 2 θ ,
λ f = ( λ / n ) 2 sin ( θ / 2 ) .
I = d 2 R e i K · R Δ ρ ( R ) = 2 π Δ ρ ( K ) .
E 2 tot ( R , t ) = E 0 2 I 2 = 4 π 2 E 0 2 Δ ρ ( K ) 2 .
I 2 = j k e i K · ( R j - R k ) .
I 2 A d 2 R d 2 R e i K · ( R - R ) × j k δ ( R - R j ) δ ( R - R k ) .
j k δ ( R - R j ) δ ( R - R k ) = l = 1 N δ ( R - R l ) δ ( R - R l ) + j k j , k δ ( R - R j ) δ ( R - R k ) .
j k δ ( R - R j ) δ ( R - R k ) = l = 1 N δ ( R - R l ) δ ( R - R l ) + N ρ ( R R 1 ) δ ( R - R 1 ) .
I 2 = N + N A d 2 R A d 2 R e i K · ( R - R ) × δ ( R - R 1 ) ρ ( R R 1 )
I 2 = N + N A d 2 R e i K · ( R 1 - R ) ρ ( R R 1 ) .
I 2 = N + N d 2 R e i K · R ρ ( R 0 ) .
ρ ( R 0 ) = ρ [ 1 - f ( R ) ] .
f ( R ) = 0 , R R c , f ( R ) = 1 , R R c .
I 2 = N - N A ρ f ( R ) e i K · R d 2 R + N ρ A e i K · R d 2 R .
I 2 = N ( 1 - ρ A f ( R ) e i K · R d 2 R ) .
I 2 N ( 1 - ρ A f ( R ) d 2 R ) .
I 2 N ( 1 - ρ A c ) .
ρ = N / A 1 / A 0 .
I 2 N [ 1 - ( A c / A 0 ) ] .
ρ f ( R ) e i K · R d 2 R ,
δ N 2 Ω = ( N Ω - N Ω ) 2 = N 2 Ω - N 2 Ω
= N Ω ( 1 - ρ Ω f ( R ) d 2 R ) .
I 2 = δ N 2 Ω .
E tot 2 = E b 2 + E c 2 .
E b 2 = N E 0 2 [ 1 - ρ f ( R ) e i K · R d 2 R ] .
E c 2 = α = 1 p N α 2 E 0 2 .
N α 2 = 1 p α = 1 p N α 2 .
E c 2 = p N α 2 E 0 2 .
E 2 tot = N E 0 2 ( 1 - ρ f ( R ) e i K · R d 2 R + p N N α 2 ) .
( p / N ) ( N α 2 .
= ( p N α / N ) ( N α 2 / N α ) ,
R j = R α + R α j .
E c = ( - E 0 ) e i ( k 0 R - ω 0 t ) α = 1 p e i K · R α ( j = 1 N α e i K · R α j ) .
j = 1 N α e i K · R α j = A α d 2 R e i K · R ( j = 1 N α δ ( R - R α j ) ) ,
J α ( K ) = j = 1 N α e i K · R α j = A α ρ e i K · R d 2 R .
J α ( K ) = ρ π a α 2 = N α .
E c = ( - E 0 ) e i k R - ω 0 t ( α = 1 p J α ( K ) e i K · R α ) .
E tot 2 = E b 2 + E c 2 .
E b 2 = E 0 2 I 2 ,
E c 2 = E 0 2 α = 1 p α = 1 p J α ( K ) J α ( K ) e i K · ( R α - R α ) .
E c 2 = E 0 2 α = 1 p J α ( K ) 2 .
E tot 2 = N E 0 2 [ 1 - ρ f ( R ) e i K · R d 2 R + 1 N α = 1 p J α ( K ) 2 ] .
J α ( K ) = ρ R = 0 R = a α RdR 0 2 π d φ e iKR cos φ .
J α ( K ) = 2 π ρ 0 a α RdRJ 0 ( K R ) .
J α ( K ) = ρ π a α 2 [ 2 J 1 ( K a α ) K a α ] .
J α 2 ( K ) = N α 2 [ 2 J 1 ( K a α ) K a α ] 2 .
J α 2 ( K ) ρ 2 ( π a α 2 ) 2 [ 1 - 1.12 ( K a α 3 ) 2 + 0.422 ( K a α 3 ) 4 ] 2 .
J α 2 ( K ) = ρ 2 ( π a α 2 ) 2 8 π cos 2 [ K a α - ( 3 π / 4 ) ] ( K a α ) 3 .
J α 2 ( K ) ρ 2 ( π a α 2 ) 2 4 π 1 ( K a α ) 3 .
( 1 2 π ) < d α ( λ / n ) < ( 3 2 π ) .
= 1 N α = 1 p J α ( K ) 2 .
= 1 N d a N ( a ) π a 2 ρ [ 2 J 1 ( K a ) K a ] 2 ,
N = ρ A .
P ( z ) = P 0 e - τ z .
τ = π 2 8 r 0 4 ( 2 π λ ) 3 ( m 2 - 1 ) 2 [ 1 + 2 ( m 2 + 1 ) 2 ] × ρ [ 1 - ρ f ( R ) e i K · R d 2 R + ] ,
P tr P 0 exp - { 2.5 [ 1 - ρ f ( R ) e i K · R d 2 R + ] } .
τ = 24 π 3 ξ 2 N a V a 2 / λ 4 .
ξ = n a 2 - n l 2 n a 2 + 2 n l 2 ,
N a = ( 0.3 N 0 / M a ) ζ .
V a = v ¯ M a / N 0 .
τ = 24 π 3 λ 4 ( 0.3 ) v ¯ 2 N 0 ξ 2 M a ζ .
τ = 0.4 × 10 - 7 M a cm - 1 .

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