The importance of the cornea and the peculiar circumstances of its transparency have stimulated research for many years, but despite its apparent suitability there have not been any adequate quantitative investigations by means of polarized light. We shall show in this and the following paper that this technique, besides yielding results which throw light on the ultrastructure and fibre arrangement of the tissue (Naylor, 1952), also affords a means of assessing changes in the intra-ocular pressure of the intact eye by quantitative measurements of the photo-elastic properties of the cornea.
The application of quantitative polarization techniques to biological tissues presents many difficulties; nevertheless, measurements can be carried out with great accuracy if the tissues can be suitably examined in a polarizing microscope (Swann & Mitchison, 1950). In the present case the ordinary microscope was of limited value, since individual corneal fibres cannot easily be isolated and changes in their birefringence with changing intra-ocular pressure can only be measured in the intact eye. Modifications of the usual techniques of measuring retardation were therefore required, and these may be of some interest to those concerned with measurements in biological tissues elsewhere.
A study of the corneal interference figure, and measurements of the retardation of the cornea suspended in a pressure chamber under near physiological conditions have already been described (Stanworth & Naylor, 1950). This study—in particular the finding that the retardation was essentially zero for light incident normally to the surface—led us to postulate that the cornea behaves as a curved uniaxial crystal plate with its optic axis perpendicular to its surface at all points, and that this could be explained qualitatively on the basis of a random orientation of the corneal lamellae. It is possible to show this quantitatively, and to deduce from the measurements the value of the birefringence of the individual corneal fibres.
From the present point of view the cornea consists of about 100 curved layers, each containing roughly parallel fibres. Each fibre has its optic (slow) axis along its length (His, 1856), and each layer, therefore, behaves as a crystal plate with its optic axis in the plane of the plate.
dΦ is the phase of the vibration along the major axis of the emergent elliptical vibration, with respect to the incident light,
dΩ is the angle between the major axes of the incident and emergent light,
dU is the ellipticity of the emergent light,
u is the ellipticity of the incident light,
du is the ellipticity of the light emerging from any one plate,
o and e are the times taken by the ordinary and extraordinary vibrations respectively to traverse any one plate,
γ is the angle between one plate and the preceding one, being positive if measured clockwise and negative if anticlockwise,
ϕ is the retardation produced by any one plate,
Σ indicates the sum of all the terms analogous to that written and in which the quantities have the values which apply successively to each of the plates.
The same result is obtained by the deduction from the above formulae that, under these conditions, the order of the plates is immaterial ; the total effects of the randomly orientated plates is the same as that of a series in which the plates are arranged in pairs with their axes mutually perpendicular. In a pair of such plates, the vibration which passes through the first plate at the speed of the ordinary vibration passes through the second plate at the speed of the extraordinary vibration and vice versa; if the plates are of equal retardation there is no retardation of one vibration with respect to the other, and the direction perpendicular to the plates is therefore an optic axis.
The retardation for light passing in any other direction can be calculated from the velocity ellipse (Fig. 1). Let OA represent the optic axis of the first plate, the magnitude OA = a being the velocity of the ordinary wave, and OB = b being the extreme value of the velocity of the extraordinary wave. Then for light travelling in any other direction OP, making an angle (θ) to the optic axis, the ordinary wave travels at velocity a, vibrating perpendicular to the plane of the paper, and the extraordinary wave travels at velocity OP=p, vibrating in the plane of the paper. If the optic axis of the next plate lies along OC, then the ordinary wave travels at velocity a, vibrating in the plane of the paper, and the extraordinary wave travels at velocity b, vibrating perpendicular to the plane of the paper.
For vibrations in the plane of the paper, the time taken to traverse the system is, therefore, l/p + l/a, where l is the path length in one lamella.
For vibrations perpendicular to the plane of the paper, the time taken to traverse the system is l/a + l/b. The path difference per unit path length is , and the effective birefringence is . As OP approaches OA, µp approaches µa and the effective birefringence approaches . The system therefore behaves like a single plate of birefringence .
If the corneal fibres are randomly arranged, then the cornea considered as a ‘plate’ will have a birefringence half that of the constituent fibres.
The birefringence (ω) of the corneal fibres can then be calculated easily from measurements of the retardation of the cornea for light passing parallel to the axis of the symmetry. Since the effective birefringence for light passing at an angle a to the optic axis is proportional to sin2α (Bear & Schmitt, 1936) the retardation A for this light path is given by d ω sin α tan α, where d is the corneal thickness.
Assuming that the refractive index of the cornea is 1·376, the radius of curvature of the anterior surface is 8 mm., and the thickness increases from 0·5 mm., in the centre to 0·9 mm. in the periphery (Steindorff, 1947), the value of angle a, and hence the expected retardation can be calculated for a light path entering the cornea at any point (Table 1). It will be seen that if the birefringence of the corneal fibres is 0·0028, the calculated retardation is in good agreement with the observed retardation.
It is shown mathematically that the cornea, in which the fibres are randomly arranged, has a coefficient of birefringence equal to half that of its constituent fibres. Analysis of previously published results indicates that the latter value is 0.0028.