I. Introduction

In 1957, Maurice produced a pioneering analysis of the transparency of the corneal stroma.[1] Electron microscope photographs[2],[3] had shown that the stroma consists of long fibersof collagen in a mucopolysaccharide ground substance. Maurice calculated the scattering of lightproduced by each collagen fiber. He was able to show that, if each fiber radiatedindependently of the others, a little more than 90% of the incident lightwould be scattered. Such a cornea would be opaque. Maurice rightly concluded that to understand thetransparency it was necessary to take into account the correlation in the phases of the wavesscattered from each fiber.

Perhaps the simplest and most striking demonstration of the effects of phase correlation inscattering is found in the diffraction of light and x rays from perfectly regular gratings orlattices. In such arrays the position of each scatterer is known exactly relative to all the others,and it is simple to calculate the summation of the phases of the scattered waves. Calculation showsthat the light scattered from a perfectly regular lattice is essentially zero in all directionsexcept those few which correspond to Bragg reflections. It was therefore natural for Maurice tosuggest that a similar perfect regularity existed in the corneal stroma: “For a tissue to betransparent it is necessary that its fibrils are parallel, equal in diameter and have their axesdisposed in a lattice.”[1]

Despite the mathematical appeal of the perfect lattice, it is not necessary that the stromalfibers be arranged in a lattice to obtain transparency. Lattice regularity neither is requiredtheoretically nor is it found experimentally. The Bowman’s zone of the shark is a strikingexperimental case in point. Here the collagen fibers are arranged in apparently complete disorder,their axes being randomly oriented in every direction. Goldman and Benedek observed that thethickness of this zone in the shark is so great that if these fibers were treated as independentscatterers the shark cornea would be opaque.[4] Clearly, even inthis disordered array, the phase relations between the waves scattered from each fiber reducessubstantially the intensity of the scattered light. To understand this theoretically, Goldman andBenedek pointed out that the collagen fibers in both human and shark stroma are spaced by distancessmall compared to the wavelength of light. As a result there is considerable correlation between thephases of light waves scattered by neighboring fibers, and this will produce the required reductionin the scattered intensity. They went on to point out that the theory of the scattering of lightfrom random arrays shows that large scattering and consequent opacity results only if there aresubstantial fluctuations in the index of refraction which take place over distances comparable to orlarger than the wavelength of light.[4] In a later experimentalinvestigation they showed that opaque corneas did in fact contain irregularities in the density ofthe collagen fibers and so-called lakes where there was no collagen. The dimensions of thoseirregularities and lakes were indeed comparable to or larger than the light wavelength.[5] Our assertion that lattice structure is not essential fortransparency has been investigated in detail in two recent papers by Feuk[6] and by Hart and Farrell.[7] The formerauthor observed that the arrangement of stromal fibers was pictorially consistent with randomdisplacements of the fibers around perfect lattice positions. The rms displacement found to bequalitatively consistent with the electron micrographs was about one tenth of the average interfiberspacing. Feuk then used the Debye-Waller theory of thermal diffuse x-ray scattering to calculate thescattering of light from this arrangement of fibers and found that the theoretical magnitude of thescattering was consistent with the observed scattering from normal stroma. While this approachprovides a useful semiquantitative analysis of the scattering, it suffers from the defect of notproperly including the effects of correlation in position between pairs of neighboring fibers. It isimportant to include this correlation to calculate correctly the wavelength dependence of thescattering.

Hart and Farrell alternatively computed the detailed probability distribution function for therelative position of fibers from photographs of corneal stroma. This distribution function showedthat the position of pairs of collagen fibers remained correlated only over two near neighbors atmost. This is very different from a perfect lattice. They were able to show, by a precisemathematical summation of the phases of the waves scattered by such a partially ordered ariay, thatboth the magnitude and the wavelength dependence of the scattered light was in good agreement withthat found experimentally.

In this paper we present:

As this subject is of great interest to the research ophthalmologist, we have endeavored topresent the theory in its simplest mathematical form. We hope by this means and by carefulpresentation of the concepts involved to bring out clearly the subtle physical considerations thatlead to transparency or opacity.

II. Scattering of Light from a Two-Dimensional Array

A. Normal Corneal Stroma

Electron microscope photographs show that the corneal stroma consists of lamellae within whichcollagen fibers are laid down approximately parallel to each other in a mucopolysaccharide groundsubstance. We shall consider the scattering of light from a single lamella. The scattering effect ofmany lamella follows simply from that of a single one. Let the fibers be arranged so that their axesare parallel to the z direction, as shown in Fig.1. In Fig. 1, 0 is the arbitrary origin of thexyz coordinate system. Then the vector R_jfrom this origin to the intersection of each fiber with the xy plane will specifythe position of each fiber. In general, regardless of the orientation of the fibers, or the locationof the field point, the fiber positions can be specified completely in terms of the two-dimensionalposition vectors R_j which specify the points ofintersection of the fibers in that plane which includes the observation point and is normal to thefiber axis.

The incident light wave has the formE₀e^{i(k₀·r−ω₀t)};k₀ is the wave vector of the incident wave in the medium. The magnitude ofk₀ is 2π/(λ/n), whereλ is the wavelength of the light in vacuum and n is the mean index ofrefraction. ω₀ =2πν₀ is the angular frequency of the light wave. We usehere the complex exponential as a mathematically convenient means of expressing the sinusoidaltraveling light wave. As this light wave falls on the fiber–ground substance combination,the oscillating electric field in the light wave induces at each point in the medium an oscillatingelectric dipole moment whose size depends on the square of the index of refraction at that point.These oscillating dipoles then each reradiate a new electric field in all directions in space. Allthese radiated wavelets add together to produce both the transmitted light beam and the sidewaysscattered light. The scattered light removes energy from the incident beam and thereby decreases theintensity of the transmitted beam. Thus, because of the sideways scattering, the transmitted beam isweaker than the incident beam. Consequently, we find that the scattering medium is not perfectlytransparent. The calculation of the turbidity of a medium therefore requires an evaluation of theamount of light it scatters.

If the stromal collagen fibers had an index of refraction(n_c) that was the same as that of the ground substance(n_i) there would be no net scattered field. The sum ofall the fields radiated from each point in the medium would add up to zero in all directions exceptin the direction of the transmitted beam. The scattering of light can occur only if there is adifference between the index of refraction of the collagen and the ground substance. Insofar as thescattering of light is concerned, we may regard the corneal stroma as being made up of fibers eachof which produces a scattered field proportional to(n_c² −n_i²).

1. The Field Scattered by Each Collagen Fiber

The electric field radiated from a single collagen fiber located at the positionR_j contains two parts: an amplitude part and a phase part.The amplitude part gives the size of the radiated field. The phase part tells precisely where thisscattered field is in its oscillation cycle. This phase factor crucially determines the scatteringprocess, because, as we add together the effect of many scatterers, the net result depends in detailon the summation of phases of the waves scattered from each fiber.

If a collagen fiber is at the point R_j, the electricfield that it scatters to the field point at the position ℛ (see Fig. 2) is given by

In the succeeding paragraphs we shall discuss and define each of the factors that enters intoEq. (1).

We assume that ℛ ≫R_j. The amplitude factor isE′_0j, and the phase factor ise^{i(k₀−k)·R_j}.The factore^{i(k₀ℛ−ω₀t)}is the same for all the fibers since ℛ is the distance from the origin tothe field point. This factor represents the fact that the radiated field is an electromagneticdisturbance with the same wavelength and frequency as that of the exciting field.

In subsequent analysis the amplitude factor E′₀ will not beconsequential. Nevertheless for the sake of completeness we write below the magnitude and directionof E′_0j if the incident fieldE₀ is polarized as shown in Fig. 3. Thescattered field is a function of the distance r in the xy planeand the angle θ between the x axis and the direction ofr. As indicated in Fig. 3, ris the distance in the xy plane from the axis of the fiber to the observation orfield point. We break E′₀ into a component in the zdirection and a component in the xy plane in terms of the unit vectors ${\widehat{1}}_z$ and ${\widehat{1}}_{\theta }$. In the case that the radius of the fiberr₀ is small compared to the wavelength of light λ in the mediumoutside the fiber, and if all the fibers are within a region small compared to the distanceℛ to the field point, thenE′_0j is independent of the positionR_j of fiber and is given by

where m =n_c/n_i.This result[7],[8] statesthat when the incident field is polarized along z, the scattered field is polarizedentirely along the z direction and is independent of θ.However, when the incident field is polarized along y, the field is in thexy plane in the direction of ${\widehat{1}}_{\theta }$ and its amplitude varies as cosθ in accordancewith dipole radiation. The amplitude of the scattering from each fiber is proportional to the squareof the fiber radius and is inversely proportional to $r^{\frac{1}{2}}$ because each fiber is regarded as an infinite cylinder. Because of thecylindrical symmetry the radiated field propagates only in the plane of the vectorR_j. In Figs. 1–3 this plane is thexy plane.

The factore^{i(k₀−k)·R_j}in Eq. (1) merits careful consideration. Here,k₀ is the wave vector of the incident light, and k is the wavevector of the scattered light wave. This exponential represents the effect on the scattered field ofthe difference in path between rays scattered from the fiber atR_j and a fiber at O. We can see this in Fig. 4. The ray emanating from O and moving in the scattered directionθ travels a distance R_jcosθ′ further in its path to the observer of the scattered fieldthan the ray scattered from j. In units of the optical phase this difference is[2π/(λ/n)](R_jcosθ′), where λ/n is the wavelength oflight in the stroma. It must be observed, however, that the oscillation induced by the incidentfield at the point R_j took place at a time later than thatat O. This introduces a factor(2π/λ/n)R_jcosφ to describe this retardation. The net difference in optical phasebetween rays scattered from O and j then is

It is convenient to express this in terms of the wave vectors k₀ andk. k₀ is a vector pointing in the propagation direction of the incidentlight and having length 2π(λ/n), whereask is a vector of the same length but pointing in the direction of the scattered wave.Using the dot product notation, viz.,

we see that we may write the phase difference ΔΦ in terms of the differencevector k − k₀ = K. HereK is called the scattering vector:

The scattering vector K will enter our discussion continually, so let us examine itsproperties at this point. Since K is the difference between k₀and k, we see from Fig. 5 that its magnitudeis

This figure also shows clearly that the direction of K is perpendicular to the planebisecting k₀ and k.

2. The Field and the Intensity of Light Scattered from Many Fibers

We now calculate the total electric field and the total power scattered by all the fibers. Thetotal electric field as observed at some field point is the sum of the electric fields radiated byeach fiber. If the observation distance ℛ is large compared to the cornealthickness then E′_0j is independentof fiber position (j), so that if there are a total of Nilluminated fibers we have the following expression for the total scattered fieldE_tot(ℛ,t):

or

The last term, which is the sum of the phase factors of each of the scattered fields can becalled the interference function I:

Since, as we have pointed out earlier, the R_j vectorsare in a plane, the interference function is a sum over positions in a plane. Equation (9) shows very clearly that the total scattered field isdetermined by the summation of many waves at a particular point in space. Each of the constituentwaves has a different phase from the others. The summation of waves is illustrated schematically inFig. 6. The interference function I describesthe total sum of all the waves adding together, some with positive values, some with negativevalues. It is an interference function because the net result of all the waves is determined by thetotal effect of many constructive and destructive interferences between the constituent waves. Thescattering of light waves from collagen fibers is perfectly analogous to the superposition of manywaves on the surface of a ripple tank. In such a tank, surface waves are generated at one side ofthe tank by two or more ripple generators. At every point in the tank the net displacement of thesurface is fixed by the superposition of the surface waves. In some regions at a particular momentthe waves interfere constructively to produce a large amplitude disturbance. At other points thewaves destructing interfere and give no net displacement. The light waves superpose in a similarway. The result at any point in space is determined by the superposition of the phases of all thewaves that exist at that point. The interference function therefore is at the very heart of thecalculation of the scattered field. In the discussion that follows we show how a carefulconsideration of this function leads to an understanding of the transparency and the opacity of thecornea.

Let us write the interference function of Eq.(10) in an alternative form. For convenience we introduce the well known delta functionδ(R −R_j) which has the property that its integral

is zero if the area of integration does not include R_j,and the integral is unity if the area does include the pointR_j. In terms of these delta functions, which in effectlocate the positions of the fibers, we may write I as an integral over a sum ofdelta functions, viz.,

Here A is the total illuminated area of the lamella and N isthe total number of fibers in the illuminated area.

Now, if the number of fibers in the scattering region is sufficiently large that there are manysuch particles in each area of dimension (1/K) × (1/K),then, since the factore^iK·Ris constant in each such small area, we may describe the sum of delta functions as a local numberdensity ρ(R). Thus we write

This density ρ has some average value〈ρ〉 which is the same at all points in space. However, asone moves from one region to another the density of fibers will fluctuate around this average value,so that we should write

where Δρ(R) is the deviation of the density in theregion around R from the average density 〈ρ〉= (N/A). If we put Eq. (13) into Eq.(11) we obtain

The first integral has the property that it is equal to zero, if A is largecompared to the wavelength of light, for all values of K except for K= 0. Thus the uniform part of the density produces no scattering of light to the side. Thescattering light is produced entirely by fluctuations in the density, and the magnitude of thescattering grows larger as the fluctuations in density grow larger. We may carry this observationfurther by recognizing that the final term in Eq.(14) is in fact just the fourier component of the density fluctuation whose wave vector isequal to the scattering vector K. The meaning of this remark may be understood in thefollowing way. In Fig. 7 is plotted schematically thefluctuation in the density as one moves in some direction in the scattering medium. This randomfluctuation in the density can be looked upon as being made up of many perfectly sinusoidal waves ofwave vector q, each of which has some amplitudeΔρ(q) in the following way:

This is the mathematical statement that Δρ(R) is asum of sinusoidal wavese^iq·Reach of which has amplitude Δρ(q). This amplitudeΔρ(q) is determined byΔρ(R) by inverting Eq. (15), namely

This equation is very similar to Eq. (14). Infact the final term in Eq. (14) is exactly equal toΔρ(q′) when q′ =K. Thus, of all the fourier components present in a sinusoidal decomposition of thedensity fluctuations only that which has a periodicity or a wavelength(λ_f) equal to 2π/K isresponsible for the scattering of light in the direction θ. Since

we may write

This equation is nothing other than the famous Bragg reflection condition. In the present contextit can be understood to mean that light is scattered an angle θ if thereexists in the scattering medium a fluctuation in density or index of refraction whose wavelength isgiven by [λ/2n sin(θ/2)]. Thesmallest such wavelength is one-half of the wavelength of light in the medium. It is on the basis ofthis very general consideration (which also applies in three dimensions) that we have suggestedpreviously[4],[5] that thescattering of light is produced by those fluctuations in density whose spatial dimensions arecomparable to or larger than one half of the wavelength of light. Electron microscope photographsresolve much smaller structures, of course. In examining such photographs for the source of thetransparency or opacity of ocular media, one must search for those structural features that containfourier components whose wavelengths are comparable to or larger than the wavelength of light.

In the discussion which follows, I shall show why the fourier amplitudeΔρ(K) is small in normal corneal stroma. Thediscussion given above shows that

The intensity of the scattered light is proportional to the square of the scattered electricfield. Using Eqs. (9) and (10) the squared electric field is

To calculate |I|² it would appear that we would need to know indetail the precise distribution of fibers in the lamella. This we do not know. The difficulty can becircumvented as follows. Suppose we imagine that we scatter light not from a single lamella but froma succession or an ensemble of many lamella. The intensity of the light scattered from each lamellawill be different, but all the scattered intensities will fluctuate around an average value.Similarly, the square of the interference function will fluctuate around some average〈|I|²〉. It is this ensemble average, denoted by〈〉, which measures the magnitude of the scattered intensity, and it is this averagethat we will calculate. It should be realized that in the corneal stroma such an average occurs asthe light is scattered from many lamellae. Thus the scattered light intensity is proportional to theproduct〈|E′₀|²〉〈|I|²〉.The first average is taken over all orientations of fibers relative to the incident lightpolarization. The second average is taken over the distribution of fiber positions. The greatadvantage of the ensemble average is that it does not require knowledge of the detailed microscopicproperties of a particular lamella but depends only on a statistical or probabilistic description ofthe fiber arrangements.

We can now calculate 〈|I|²〉 in the following way.From Eq. (10) we have

Introducing the delta function representation again, this can be expressed as

The ensemble average of the double sum can be decomposed as follows:

N is the total number of fibers illuminated. The first term on the right-handside of Eq. (23) comes from the j= k term, the second from the j ≠k term. The second term in Eq. (23)is an ensemble average over N rows of (N − 1) terms. Theterms in the row with j = 1 are δ(R− R₁)δ(R′ −R₂) + δ(R −R₁)δ(R′ −R₃) + … δ(R −R₁)δ(R′ −R_N). The ensemble average over each row gives the sameresult, so that the sum is N times as big as the sum over a single row. Theensemble average over a single row with j = 1 isδ(R − R₁)[ρ(R′|R₁)].Here ρ(R′|R₁) may be calledthe conditional number density distribution.ρ(R′|R₁)d²R′is the average number of particles within aread²R′ around R′, underthe condition that there is certainly a particle at the position R₁. Thus wemay write

On putting this into Eq. (22) we find using theproperties of the delta function that

or

Since the conditional number densityρ(R′|R₁) is a functiononly of the distance |R′ − R₁| we candefine a new variable R″ = R₁ −R′ in terms of which Eq.(26) takes the form

In Maurice’s analyses of the scattering of light he first considered the scatteringassuming that each fiber scattered independently of the others. In terms of thepresent analysis this means that the second term on the right side of Eq. (27) is assumed equal to zero. Maurice found that under such anassumption the total scattered intensity would be so great that the cornea would certainly beopaque. It is not opaque because the second term in Eq.(27) which describes the correlation between the phases of waves scattered from nearby pairsof particles nearly cancels the N term. Maurice’s thesis was that for sucha cancellation to take place, the distribution of neighboring fibers had to be that of a perfectlattice. While such a distribution can indeed produce the required cancellation, it is by no meansnecessary that such an extreme correlation occur. We now show that even with therelatively poor correlation that exists in the position of pairs of fibers in the cornea that thesecond term in Eq. (27) can nearly cancel the first.To do this, let us examine the properties ofρ(R″|0). For small values ofR″, ρ(R″|0) must go tozero because a second fiber cannot be closer to the fiber at 0 than a distance equal to the fiberdiameter. In fact, in the cornea ρ(R″|0) is zerofor larger values of R″ than the fiber diameter. For values ofR″ substantially greater than some correlation rangeR_c the number of particles ind²R″ will be completely unaffected by thepresence of the particle at the origin and ρ(R″|0)will become equal to the average density 〈ρ〉 =(N/A). In Fig. 8 we plot thegeneral form of ρ(R″|0). Wigglesoccur in ρ(R″|0) above and below the value of〈ρ〉 if the neighbors are located in fairly well-definedpositions relative to a particle at the origin. The first positive bump in Fig. 8 corresponds to the location of near neighbors, the small negative bumpis the space before the next position of second nearest neighbors. In the schematic diagram shownbelow there is no correlation over more than, say, the distance to second nearest neighbors. Theactual form of ρ(R″|0) has been constructed byHart and Farrell[7] from electron microscope photographs offiber position in normal corneas. Their function has the form shown in Fig. 8 and demonstrates clearly that after about two near neighbors there is no correlationin the position of particles. Nonetheless the required reduction in scattering takes place. We cansee why this occurs. Because of the form of ρ(R″|0) wecan write it as

Here

This function f(R″) is plotted schematically in Fig. 9. Putting Eq.(28) into Eq. (27) we find

The last term is identically zero if the illuminated area A is much larger thanthe wavelength of light unless K = 0. Thus for scattering in any directionother than the forward direction we have

This equation tells us precisely how the correlation in the relative position of particles, asexpressed in the function f(R″), reduces the scatteringdepending on the size of the second term in the brackets in Eq. (30) compared to unity. It is possible at the outset to make an estimate of the size ofthis term. Let us suppose for convenience that f(R″) goesto zero for values of R″ so much smaller than the wavelength of light thatthe integral can be evaluated assuming thate^iK·Rchanges very little from unity. If this condition applies, then we can write

The integral∫f(R″)dR″ isan effective correlation area A_c. Sincef(R″) ~ 1 up to the correlation distanceR_c, the integral is equal approximately to thecorrelation area (A_c) over which there existssubstantial correlation in the position of particles. Thus we may write Eq. (31) simply as

Now, we must recognize that 〈ρ〉 is the number ofparticles per unit area

Calling (A/N) the average available area perparticle (A₀) we see that

Clearly, if the scattering particles have a small area of correlation (this is of the same orderas the size of the particle itself) in comparison to the area, on average, available to theparticle, the factor A_c/A₀will be small compared to unity and the scattering proportional to the total number of particlesN. On the other hand, when the correlation area is about equal to the areaavailable to the particle, i.e., when the particles become closely packed,A_c/A₀ becomes close tounity and as a result |I|² is reduced very markedly belowN.

In three-dimensional systems the factorA_c/A₀ is replaced byV_c/V₀, the ratio of thecorrelation volume to the available volume. For gases at low densityV_c/V₀ ~ 1/1000 and thescattered light intensity is N times the scattering produced by a single particle.However, in a liquid V_c/V₀is of the order of unity and the scattering per particle is much smaller than that produced by anisolated particle.

In the cornea the correlation effect represented by the factorA_c/A₀ is very important.This can be seen at the very outset since A_c andA₀ can be estimated from electron microscope photographs of normalstroma. A₀ = 1/〈ρ〉 isapproximately 0.3 × 10⁶(Ų).[1] We may estimate A_c by noting that the averagespacing of collagen fibers is 550 Å and that the centers of two fibers are never closer thanabout 300 Å. Taking the correlation range R_c asabout 400 Å, we see thatπR_c² ~A_c estimated in this crude manner is 0.5 ×10⁶(Ų). Thus we crudely estimate from the outset thatA_c/A₀ ≃ 1.7. Thisnumber is of the order of unity so that it is immediately obvious that the correlation in positionsof fibers plays a very important role in the determination of the transparency. To establish moreaccurately the effect of this correlation we must return to Eq. (31) and evaluate more accurately the correlation areaA_c by computing the integral∫f(R″)dR″.

Hart and Farrell have constructed the form of the correlation functionρ(R″|0) by making a statistical computation of thedensity of fibers around any given starting fiber. These authors have kindly informed me that theintegral〈ρ〉∫f(R″)dr″has a value between 0.8 and 0.95.[9] Thus the actual scatteringfrom the cornea is between 0.2 and 0.05 times smaller than the estimate of ~90% computed byMaurice on the basis of independent scatterers. It is important to point out that this reductiontakes place even though the particles are correlated by distances no greater than the spacing of twoneighbors.

Hart and Farrell have also computed[9] the quantity

which appears in the accurate expression for 〈|I|²〉in Eq. (30) and includes the phase factore^iK·R″.In the cornea R_c is large enough so that for backwardscattering KR_c ~ 0.5. Thus the exponential factore^iK·R″does change somewhat during the integration. Their calculations take this into account. They havekindly informed me that, when this factor is included, for a light wavelength of 6000 Å thisintegral has the value 0.87.[9] It is to be observed that thisaccurate result is quite consistent with the cruder estimate given above. The scattering in effectis reduced to a value eight times smaller than that produced by independent scatterers because ofthe correlation between the position of the fibers.

There is yet another way to understand the marked reduction in the intensity of scattering belowthat which obtains when the particles are treated as independent scatterers. This is to observe thatthe mean square fluctuation〈δN²_Ω〉 in the number ofparticles in some area Ω, which is arbitrary (but must be large compared with thecorrelation area A_c ≃πR_c²) can be written as

This last equation is of the same form as Eq.(31). That is, if Ω is equal to the illuminated area, and if the range of thecorrelation R_c is small compared to2π divided by the wavelength of light, thene^iK·R″can be replaced by unity in Eq. (30), and Eq. (31) results. Under these conditions〈|I|²〉, the mean square value of the interferencefunction is equal to the mean square fluctuation in the number of particles in the area ofillumination:

Equation (36) states that the mean squarefluctuation in the number of particles in any chosen area Ω is proportional to the averagenumber of particles 〈N_Ω〉 in that area. Theconstant of proportionality (1 −A_c/A₀) is much smaller thanunity, however, when the particles are densely packed in the sense that the correlation areaA_c is about as large as the available areaA₀. This is the case in the cornea. There appears to be an interactionbetween collagen fibers which keeps them from coming very close to one another. This raisesA_c to such a value thatA_c/A₀ ~ 1. In terms offluctuation in the number of fibers, the collagen fibers act as if they were nearly close-paced.Under these conditions, of course, the fluctuation in the number of fibers in any area will be smallbecause a considerable expenditure of energy would be required to change the number in that area.The reduction in the scattering thus can be interpreted as resulting from the fact that the collagenis effectively densely packed, and as a result fluctuations in the density of these fibers aresuppressed to values much smaller than that appropriate to a random arrangement of pointparticles.

B. Swollen, Opaque Corneas

Having now discussed the delicate conditions required for the transparency of the corneal stroma,let us give an analysis of the scattering of light produced by the microstructural alterationsobserved in swollen corneas. Electron microscope photographs, like that shown in Fig. 10, show that swollen corneal stroma contains irregular regions or lakesin which there is no collagen present at all. These lakes are irregular in cross section and aretaken to have the form of long cylinders. Such regions represent places where there is a largefluctuation in the density of collagen. As we have seen in the previous discussion, the scatteringof light is produced by fluctuations in the fiber density or, equivalently, by fluctuations in theindex of refraction. Hence we can expect such lakes to scatter light.

In Fig. 11(a) we plot how the index of refraction of thestroma varies as one moves along some particular direction in a swollen cornea containing lakes. Theindex of refraction (n) shows sharp bumps at the position of the fibers by anamount equal to the difference between the index of the fibers(n_c) and the index(n_i) of the mucopolysaccharide ground substance.[10] The spatial variation in index of refraction shown in Fig. 11(a) produces a scattered field which can be regarded asoriginating as the sum of two arrangements of scattering sources. In Fig. 11, we show the scattering amplitudes which correspond to the two arrangements. Thefirst arrangement, (b) in Fig. 11, is the distribution offibers which would have occurred had the lakes not been present. The second arrangement (c) consistsof fibers, each of which has a negative scattering amplitude, located at positions in the lakeswhere the fibers in arrangement (b) were placed. The scattered field from (b) plus (c) will be thesame as that from that of Fig. 11(a), the stroma with lakes.Using the decomposition indicated in Figs. 11(b) and 11(c) wecan now estimate simply the scattering produced by the swollen stroma.

1. Elementary Analysis of the Scattering from Small Lakes

The mean intensity of the light scattered by (b) and (c) is the sum of the mean intensity oflight scattered by configuration (b) plus that scattered by (c). That is,

(We shall see in 2, below, the reason why〈E_bE_c〉can be neglected.) We calculated previously the value of〈E_a²〉 in Eq. (30) and found

〈E_c²〉 can be estimated inthe following simple way. Each fiber in configuration (c) scatters a field whose amplitude is(−E′₀). If the size of each lake is smaller than, or ofthe same order as the light wavelength, each fiber radiates a field coherently with all the others.That is, the factore^iK·Ris about the same for all the fibers in a lake. As a result the fields scattered from each are inphase, and the total scattered field from each lake is(−N_αE′₀),where N_α is the number of fibers missing in thelake labeled with the index α. Since we assume that the positions of thelakes are completely uncorrelated with one another, each lake radiates independently and the totalscattered intensity is the sum of the intensities scattered by each lake. Thus

Here, p is the total number of lakes in the illuminated region. We mayconveniently define the mean square value of the number of particles per lake as〈N_α²〉, where

Thus

Thus the intensity scattered by configuration (b) is proportional to the number of lakes and tothe mean square number of missing particles per lake. This scattering can be veryeffective in reducing transparency, as we now demonstrate. Adding Eqs. (42) and (39)together we find using Eq. (38) that

It is clear that if the factor(p/N)〈N_α²〉is unity or larger, the cornea will be opaque. For in this case〈E²_tot〉 ≃NE′₀², which we know from Maurice’scalculation leads to about 93% loss of light intensity. We thus have a convenient criterionfor effectiveness of the scattering from the lakes, i.e., if we define the quantity∊ as

Then if ∊ is about unity or larger, the cornea will be opaque. On theother hand, if(p/N)〈N_α²〉is small compared to unity, the lakes will not scatter much light. It is possible to estimate∊ by means of an analysis of the electron microscope photographs.Writing

we recognize that the first factor in Eq. (45) isthe fractional number of collagen fibers that are missing in the stroma. Alternatively, this firstbracket(p〈N_α〉/N)is equal to the ratio of the area of the lakes to the area of the stroma in the illuminated area.Cursory examination of Fig. 10 suggests that for this corneap〈N_α〉/N~ 0.1. The quantity〈N_α²〉/〈N_α〉is crudely equal to the mean number of fibers missing in the average lake. This is easily twenty toforty for the lakes in Fig. 10. Thus∊ is indeed of the order of or greater than unity for that cornea.

The analysis we have just given also can be used to show why lakes whose characteristic dimensionis small compared to the wavelength of light are ineffective scatterers compared with those whosesize is comparable to the light wavelength. This conclusion flows from the fact that the scatteringis proportional to N_α², the squareof the number of particles in the lake. This is proportional to the square of the area of the lakeor to the fourth power of the linear size of the lake. Thus, the light scatteredfrom a lake of diameter say one fourth of the wavelength of light is sixteen times smaller than thescattering from a lake of diameter equal to one half of the wavelength of light. This analysisdemonstrates the action of the principle which we have expressed earlier in this paper and haveemphasized in earlier experimental work,[5] namely thatirregularities in the ocular media whose linear dimensions are comparable to or larger than thewavelength of light are most effective in scattering light incident upon them.

As the size of the lakes grows large in comparison to the light wavelength the scattering fromthe lakes will continue to grow, but not so strongly as the square of the number of fibers in eachlake. This occurs because the waves scattered from each of the fibers in the lakes in arrangement(b) are not in phase with one another. Hence there is a cancellation between the radiated fieldswithin a lake, and the scattered field no longer is proportional to the number〈N_α〉 of fibers ineach lake. Of course, if the size of the lakes becomes very large compared to the wavelength oflight one may use geometrical optics to determine the reflection and refraction from theirregularities.

2. Quantitative Analysis of Scattering from Lakes

We now give a quantitative analysis of the scattering from the lakes which permits a morerigorous determination of the scattered intensity even for lakes large compared to the lightwavelength. To calculate E_c we return to the basicequation, Eq. (9), for the field scattered by adistribution of fibers. In the present case the distribution is that given in Fig. 11 (c), and the scattered field amplitudeE′₀ is the negative of that given in Eq. (2). Let the vectorR_α represent a position at or near thecenter of that lake which is labeled by index α. α= 1,2… p, where p is the total number of lakes inthe illuminated area. Let R_αj represent theposition of the jth fiber in the αth lake relative to theorigin R_α, i.e.,

Using Eq. (9) we find

N_α is the number of fibers missing from theαth lake. Using the delta function representation for the position of thefibers we see that

where A_α is the area of theαth lake. It is this area over which the integral is to be evaluated. If,on the average, the number of fibers missing in each lake is sufficiently large (say greater than~10), we may replace the sum ∑δ(R −R_α) by the average number density of fibers〈ρ〉 = (N/A). Wemay then define the quantity ${\mathcal{J}}_{\alpha }$(K) as

${\mathcal{J}}_{\alpha }$(K) is the Kth fourier component in atwo-dimensional sinusoidal decomposition of a step function which is equal to〈p〉 inside the lake area and zero outside it. If theαth lake is taken for convenience to be a circle of diameter2a_α, then we see that ifKa_α ≪ 1, ${\mathcal{J}}_{\alpha }$(K) is independent of K and equal to

Using Eq. (49) in Eq. (47) we see that

E_b is the field scattered by configuration (b) in Fig. 11. The total scattered field (E) is the sumof E_b and E_c. Thescattered intensity is, as we mentioned previously, proportional to the square of the totalscattered field. We may calculate the mean square total scattered field by making use of theensemble average. On carrying out the squaring process and ensemble averaging over all possiblelocations of lakes and arrangements of fibers the cross term〈E_aE_b〉in the ensemble average of the square may be neglected. We can understand why the〈E_bE_c〉term is negligible in the following way: E_b, the fieldscattered by the full array of collagen fibers, is proportional to the fourier transform of thefluctuation in the density Δρ(R). The mean densitymakes no contribution to E_b because, as we observedearlier, the integration is carried out over the entire illuminated region which is very largecompared to the light wavelength. On the other hand E_c,the field scattered by the substituted negative fibers in each lake, is proportional to the fouriertransform of the sum of the mean density 〈ρ〉 and thefluctuation Δρ(R) in fiber density. In the case ofthe lakes, the contribution to the fourier transform from the average density〈ρ〉 far outweighs that from the fluctuations because thefourier transform integral is computed over a region which is not very large compared to the lightwavelength. Thus, when the cross term〈E_bE_c〉is computed we see that the termE_bE_c whichis proportional to Δρ〈ρ〉will give zero when the ensemble average is carried out. The term proportional to[Δρ(R)]² will not bezero, but it will be very small compared to the〈E_c²〉 term which isproportional to the square of the fourier transform of 〈ρ〉.We therefore may neglect the cross term〈E_bE_c〉.

Thus

Now,

where 〈|I²|〉 is given by Eq. (30).〈|E_c|²〉 is given by

If the position of the various lakes R_α andR_α′ are completely uncorrelated, asappears from the electron microscope photographs, only the α =α′ terms contribute to the ensemble average and we find

Thus, the total scattered intensity is proportional to

This is of the same form as Eq. (43), except thatnow we have | ${\mathcal{J}}_{\alpha }$(K)|² instead ofN_α².

We saw in Eq. (50) that ${\mathcal{J}}_{\alpha }$(K) =N_α if the lake is small compared to thewavelength of light. We now calculate the magnitude of ${\mathcal{J}}_{\alpha }$(K) even when this is not the case. The evaluation of ${\mathcal{J}}_{\alpha }$(K) is quite straightforward as it is mathematicallyidentical to the well-known problem of the diffraction of light from a circular aperture. If weregard the lake as being a circle of radius a_α,the dimensional area differential d²R is equal toRdRdφ, where φ is the angle in the plane ofintegration between R and the direction of K. Thus

The integral over φ is the integral representation of the Besselfunction or order zero [J₀(KR)].Thus

The radial integral is also readily evaluated,[11] and wefind that

Here J₁(Ka_α) isthe Bessel function of order one. Since〈ρ〉πa_α²= N_α we see that the intensity of thescattering from each lake is given by

We plot in Fig. 12 the quantity[2J₁(Ka_α)/Ka_α]²as a function of Ka_α.

We can draw a number of useful conclusions by considering Eq. (60) and Fig. 12. First weexamine how ${{\mathcal{J}}_{\alpha }}^2$(K) varies with the size of the lake. From theasymptotic form of the Bessel functions[11] we see that forKa_α < 3

This shows that for Ka_α < 1, ${{\mathcal{J}}_{\alpha }}^2$(K) will be accurately proportional to the fourth powerof the radius of the lake. On the other hand, for large size lakes, i.e., forKa_α > 3

If we replace the cosine squared factor in this equation by one half to obtain the mean value of ${{\mathcal{J}}_{\alpha }}^2$ we find

Thus for a_α large enough so thatKa_α > 3 the scattering per lake increaseslinearly with the radius of the lake.

We may express these results in a somewhat simpler form. For small lakes (Ka< 1), ${{\mathcal{J}}_{\alpha }}^2$(K) is equal to〈ρ〉²(πa_α²)² =N_α², the square of the number offibers missing from the lake. On the other hand, if the lake is large (Ka > 3), ${{\mathcal{J}}_{\alpha }}^2$(K) is equal to(4/π)N₀²(Ka_α).Here N₀ =〈ρ〉(π/K²).If we chose the K value appropriate for backward scattering, i.e.,K = 4π/(λ/n),N₀ is the number of fibers missing from a lake of radius(λ/4πn).

The intensity of the light scattered from each lake is a continuously increasing function of thesize of the lake. In the size range 1 < Ka_α< 3, however, the dependence on size changes from a fourth power dependence on the lake radius toa linear dependence on the lake radius. Since $K{a}_{\alpha }=[4\pi /(\mathrm{\lambda }/n)]\hspace{0.17em}\text{sin}\frac{1}{2}\theta$, we see that this crossover region occurs (for backward scatteringθ = 180°) when the diameter of the laked_α =2a_α falls in the range

The crossover region in diameter occurs when d is between one half and one sixththe light wavelength (λ/n) in the cornea. Those lakes that have diameterslarge compared to this crossover size (d_α)scatter more effectively than those smaller because the scattering from the smaller lakes falls asthe fourth power of the size of the lake. Goldman’s photographs (see Fig. 12, Ref. [5]) show that lakes inopaque corneal stroma have size ~2333 Å and will therefore be very effective scatterers.

Equation (56) permits us to generalize theopacification factor (∊) first defined in Eq. (44) for lakes whose radius is small compared to the lightwavelength. For lakes whose radius is either large or small compared to the light wavelength we maydefine the quantity ∊′ as

If ∊′ is of the order of unity or even larger the cornea will beopaque, but if ∊ is less than, say, 0.1 it will be transparent. To compute∊′ from an electron microscope photograph covering some large areaA of the cornea one should first make a statistical measurement of the number oflakes [ $\mathfrak{N}$(a)da] whose radii fallbetween a and a + $\mathfrak{N}$. $\mathfrak{N}$(a) is the number distribution for lake sizes. Fromsuch a distribution one may find ∊′ by integrating over all thelake sizes in accordance with the formula

which is the representation of Eq. (65) for acontinuum of lake sizes. Equation (63) has also beenused here for ${\mathcal{J}}_{\alpha }$(K). The subscript α has beendropped in the argument of the Bessel function since the lake radius is now regarded as a continuousvariable. The quantity N is the total number of collagen fibers which would be onaverage in the area A of the electron micrograph:

3. Turbidity of the Cornea

Our discussion so far has established the condition under which light is or is not transmitted bythe cornea. It is now appropriate to describe the turbidity of the cornea more precisely in terms ofthe numerical fraction of the incident light transmitted by the cornea. It is well known that theattenuation of a beam of light, which is being scattered as it traverses the medium, follows anexponential law, namely,

Here P(z) is the power in the light beam after it has passedthrough a distance z in the medium. P₀ is the power inthe incident beam. τ is the turbidity of the medium. A large value ofτ implies great attenuation, while a small value implies littleattenuation. Maurice[1] has expressed the turbidity of thecornea (τ) in terms of the size, density, and index of refraction of thecollagen fibers under the assumption that each fiber scatters independently of the others. We caneasily adapt his result so that it includes the effect of the correlation in fiber position and thepresence of lakes. This can be done simply by recognizing that whenever the number density of fibersappears in his formula we must multiply it by the factor in brackets in Eq. (56), namely, 1 −〈ρ〉∫f(R″)e^iK·R″d²R″+ (1/N)∑| ${\mathcal{J}}_{\alpha }$(K)|². If we use for convenience thedefinition of ∊′ given in Eq. (65) we see that[1],[7]

where m =n_c/n_i,λ is the wavelength of light in the ground substance, r₀ is theradius of the collagen, and the incident light is unpolarized. On using the corneal thickness as0.046 cm, λ = 5000 Å, and Maurice’s values[1] for the various parameters in Eq.(69), one finds from Eqs. (68) and (69) that the power transmitted by the cornea(P_tr) is numerically related to the incident power by

This formula for the scattered intensity is numerically not so exact as the more complex resultsof Hart and Farrell,[7] because it does not include certainsmall effects such as the coupling between the angular dependence in the scattered field and theangular dependence of the factore^iK·R.It also does not include small corrections of the order of(k₀r₀)² in the scattering fromindividual fibers. Nevertheless, this formula gives results at 5000 Å which compare quitefavorably with the numerical values obtained by Hart and Farrell. Equation (70) shows that in the case where there is no correlation inposition of collagen fibers and no lakes [the integral in Eq. (70) is zero and ∊′ iszero], the transmitted power is exp(−2.5) ≃ 0.08 times smaller than theincident power. The correlation in fiber positions measured by Hart and Farrell reduces the exponentin Eq. (70) to the value exp 2.5(1 − 0.87)= 0.73. If the additional small but mathematically complicated effects mentioned above areincluded, the theoretical power transmitted by the cornea will be about 83% of the incidentpower. In view of experimental uncertainties about the actual turbidity of the cornea, and theaccuracy of existing estimates of the collagen and mucopolysaccharide indices of refraction,[1] this must be considered in satisfactory agreement withexperimental data.[7]

If the cornea undergoes swelling, ∊′ grows from zero, and thefactor in braces in Eq. (70) grows large. Thus, forexample, when ∊′ is 1.0, the exponent has the value exp −[2.5(1 − 0.87 + 1)] = 0.06, and only 6% of theincident light will penetrate the cornea in this case. As ∊′ growslarger with swelling to values larger than unity, the exponent describes an even greater loss intransmission. Of course, if ∊′ becomes quite large compared tounity, multiple scattering will occur and the simple exponential decrease of intensity given inEqs. (68) and (70) will not apply.

On concluding this discussion of the scattering of light by the corneal collagen fibers it isworthwhile noting that in the sclera the collagen fibers are quite large. In fact their diametersand spacings are of the same size as the wavelength of light. This produces large fourier amplitudesat wavelengths appropriate for the scattering of light, thus the sclera is opaque.

III. Theory of the Opacity of the Cataractous Lens

It is natural to inquire as to whether the theory of the scattering of light which we havepresented earlier in this paper can help us to relate microscopic alterations in the structure ofthe lens to the loss of transparency which occurs in the cataractous lens. Of course, loss oftransparency can be produced either by absorption of light or by scattering of light. In a recentreview of experimental and theoretical investigations on cataracts, Phillipson[12] has pointed out that no peaks have been found in the visible absorptionspectra of cataractous lens substances. We must therefore look to increased light scattering as theorigin of the turbidity of the cataractous lens.

The theory we have presented above indicates that the scattering of light is produced bymicroscopic fluctuations in the index of refraction. These fluctuations must be of such a size thattheir spatial fourier components have a wavelength comparable with or larger than the wavelength oflight. In the case of the lens, the absence of electron microscope photographs forces us to turn tobiochemical analyses for evidence as to the microscopic fluctuations that produce scattering in thecataractous lens.

Biochemical analyses show that cataractous lenses contain in addition to the normalα, β, and γ crystallinelens proteins,[13] an elevated percentage of insoluble proteinmaterial called the albuminoid fraction. The physical properties of this fraction are notestablished, but they are apparently proteins of large molecular weight, and they are formedpresumably by a process of aggregation or polymerization of the α,β, or γ proteins.

In the normal lens, light is of course scattered by each of the proteins in the lens. It isessential, however, to realize that, as we found in the cornea, each protein does not scatterindependently of its neighbors. The correlation in the position of pairs of proteins reduces thescattering in the lens just as the correlation between the positions of pairs of collagen fibersreduced the scattering in the cornea. In the normal lens, the fluctuation in the number of proteinmolecules over a dimension comparable to the light wavelength is small because the proteins aredensely packed, and transparency therefore results. It is to be emphasized that a“paracrystalline order”[14] of the constituentproteins is not required for transparency of the normal lens.

The presence of substantial numbers of large protein aggregates randomly distributed within thebackground of the α, β, andγ protein constituents of the lens can have an important effect on thetransparency as has been pointed out by Trokel.[14] Usingmicroradiography, Phillipson[12] has demonstrated that theopaque cortical region of lenses having galactose-induced cataracts has marked spatial fluctuationsin the protein density. If the aggregated units have an index of refraction different from that ofthe average refractive index of the lens, and if the aggregated proteins are distributed at randomthroughout the lens, then they will scatter light proportionately to their number. We can expectthat the aggregates can play a role in the lens, like that played by the lakes in the cornea. Theyproduce regions within which the index of refraction is disturbed from the average value appropriateto the lens as a whole.

We may make a quantitative calculation of the effectiveness of such aggregates if we imagine themas being spheres of index of refraction n_a imbedded ina uniform background whose index of refraction is n_l.For convenience we shall assume that the radii of such spheres is smaller than the wavelength oflight. (The analysis can easily be extended to spheres comparable to or larger than the wavelengthof light just as we extended our treatment to the case of large cylindrical lakes in the previoussection.) The problem at hand is essentially the same as the calculation of the turbidity of anaqueous solution of protein macromolecules. We may therefore use the result, well established inthat field,[15] to express the turbidityτ for unpolarized light as:

Here

N_a is the number of aggregate macromolecules percubic centimeter, V_a is the volume of each aggregatemacromolecule, and λ is the light wavelength. We can make a crude numerical estimate of thesize of the turbidity in the following way. First we observe that bothN_a and V_acan be expressed in terms of the average molecular weight of the aggregated or albuminoidmacromolecules.

Each cubic centimeter of the lens contains about 330 mg of lens protein[16]: of this, a fraction which we denote as ζ is madeup of the aggregated proteins. Thus the mass density of albuminoid fraction is 0.3ζ g/cc. In the normal lens ζ ~ 0.01, whereas inthe cataractous lens ζ can become as large as 0.20.[16] If M_a is the mean molecularweight of the albuminoid macromolecules, and N₀ is Avogadro’snumber, it follows that

We may also relate V_a to the albuminoid molecularweight by using the partial specific volume $\overline{v}$ for proteins. The quantity ( $\overline{v}$) represents the volume of solvent excluded per gram of dispersed solutemacromolecules. For most proteins $\overline{v}$ is between 0.6 cc/g and 0.7 cc/g. We may then estimate that

On inserting Eqs. (73) and (74) into Eq. (71) wefind that the turbidity is given by

Here we see that the turbidity is directly proportional to the molecular weight of the albuminoidparticles. This direct proportionality between turbidity and molecular weight is the basis of thelight scattering determination of molecular weight of macromolecules.[13] The scattering also is proportional, of course, to the fractionζ of albuminoid present and to the difference between index of refractionbetween the albuminoid and the rest of the lens, as is indicated by the factorξ² in Eqs. (75)and (72).

To calculate the turbidity exactly we must know both ξ² andM_a. At present these numbers are not well-established.We can nevertheless crudely estimate the turbidity using reasonable values of these quantities toexamine whether the present suggestion for lens opacity is tenable. If we use forn_a and n_lthe values that apply to the collagen and ground substance in the cornea we find thatξ = 0.1. These values forn_a and n_lare similar to those obtained in microradiographic studies of the distribution of proteins ingalactose cataracts.[17] We shall also use the valueξ = 0.2, which is appropriate to the cataractous lens. With thesechoices for ζ and ξ and λ = 5000Å and $\overline{v}$ = 0.6 cc/g, we find that Eq. (75) can be expressed numerically as

Since the thickness of the lens is about 0.5 cm, τz₀ ~ 2× 10⁻⁸M_a. The powertransmitted by the lens is reduced below the incident power by the factor exp(−τz₀). The lens will certainly appear turbid ifτz₀ is about equal to unity or larger. Such a value will beobtained if the albuminoid macromolecules in the lens have a molecular weight of about 50 ×10⁶ g/mole. This may not be an unreasonably large value for the molecular weight of thealbuminoid aggregates.

This crude estimate of the turbidity produced by a random distribution of protein aggregates inthe lens indicates clearly the desirability of more measurement of the physical properties of theseaggregates. In particular it would be very valuable to obtain reliable values of the index ofrefraction and the molecular weight of these protein aggregates. These data will permit a moreaccurate estimation of the lens turbidity.

Finally, if these protein aggregates should prove to be the microscopic origin of the lenscataracts, biochemical inhibition of the aggregation process would be effective in preventingcataracts.

The author acknowledges numerous stimulating discussions with Jerome N. Goldman at the RetinaFoundation, Boston. Those discussions led to the present theory of corneal transparency and opacity.I am grateful to J. Kinoshita of the Harvard Medical School who pointed out to me the remarkableincrease in albuminoid fraction in cataracts and its possible connection with lens opacity. R. W.Hart and R. A. Farrell of the Johns Hopkins Applied Physics Laboratory generously made availablenumerical results of their calculations. The author expresses his gratitude to C. Dohlman for makingpossible the collaborative relationship between the Retina Foundation and the MassachusettsInstitute of Technology. Finally, the author acknowledges with thanks the support of the Sarah ReedFund for Research on Diseases of the Eye at the Massachusetts Institute of Technology.

This research was supported in part by the Cornea Research Unit of the Retina Foundation, Boston,Mass., and by the Sarah Reed Fund for Research on Diseases of the Eye at the Massachusetts Instituteof Technology.

Figures

 

Fig. 1 Schematic diagram of arrangement of collagen fibers in a lamella. The incident light is shownpropagating along the x direction.

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Fig. 2 Schematic diagram showing the positions of the observation point (or field point)ℛ and the source point R_j forfibers in a lamella.

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Fig. 3 Direction of the polarization vector of the incident light field (E₀)and incident propagation direction (x). Also shown are the unit vectors ${\widehat{1}}_z$ and ${\widehat{1}}_{\theta }$ that are used to specify the polarization of the scattered field asobserved a distance r from the axis of a single scattering collagen.

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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 pointR_j. The scattered direction is specified by the scatteringangle θ.

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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.

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Fig. 6 The total scattered electric field at some particular field point (ℛ) 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.

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Fig. 7 Representation of the random spatial fluctuation in the fiber densityΔρ(R) as a function of position(R) inside the cornea.

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Fig. 8 General form for the conditional number densityρ(R″|0). When R″ becomesappreciably larger than the correlation range R_c, theconditional probability becomes equal to the mean density〈ρ〉. For values substantially less than the correlationrange ρ(R″|0) is zero.

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Fig. 9 General form for the function f(R″) = 1−ρ(R″|0)/〈ρ〉.

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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.

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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).

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Fig. 12 A plot of the function[2J₁(Ka_α)/(Ka_α)]²vs Ka_α.

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References

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 and G. B. Benedek, Invest. Ophthalmol. 6, 574 (1967) [PubMed]  .

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

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

7. R. W. Hart and 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 and 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 and 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 and 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]  .