1. Introduction

Corneal birefringence is a significant consideration for polarization-sensitive optical measurements in the human eye, such as scanning laser polarimetry (SLP) and polarization-sensitive optical coherence tomography (PS-OCT). For perpendicular incidence the central cornea is accurately described as a linear retarder with its slow axis oriented nasally downward (ND) [1]. The data of van Blokland and Verhelst, however, obtained in vivo with Mueller matrix ellipsometry, show that this simple description is not complete [2]. Their model treats the cornea as a curved biaxial material with its fastest axis oriented perpendicular to the corneal surface and its slowest axis coincident with the slow axis for perpendicular incidence. Götzinger, et al., using PS-OCT, failed to demonstrate biaxiality in donor corneas but did show a circularly symmetric birefringence pattern with retardance increasing toward the periphery and a circumferentially oriented slow axis, results that they explained with a simple model consisting of two birefringent layers with perpendicular optical axes [3]. Farrell, et al., developed theory to show that the optical axes of a two-layer anisotropic stack depends on the angle between the optical axes of the individual layers [4]. Their theory agrees with Götzinger, et al., when the layer axes are perpendicular, but produces a biaxial pattern for arbitrary angular separation. Recently Bone and Draper measured birefringence at the center of seven pairs of donor corneas with a petrological microscope and found all to be biaxial [5].

We felt that additional study of corneal birefringence was needed to resolve the apparent inconsistencies described above. More study was also warranted because van Blokland and Verhelst report data from only four subjects with high corneal retardance (50–80 nm), but it is now known that for perpendicular incidence the central cornea can exhibit single-pass linear retardance ranging from 0 to >100 nm [1,6]. We did this study to 1) test the curved biaxial model of corneal birefringence by another method, 2) measure subjects with a wide range of corneal retardance and 3) improve our understanding of the effect of corneal birefringence on measurements of retinal nerve fiber layer (RNFL) birefringence by SLP [7–9].

2. Curved biaxial model of corneal birefringence

A biaxial material has different refractive indexes (n) for light polarized in each of three orthogonal directions. In the biaxial model of cornea, two of these directions are tangential to the corneal surface and the third, with the smallest refractive index, is perpendicular to it [2]. The coordinate system used in the model, shown in Fig. 1(A), is aligned with these principal directions and is chosen so that n_x>n_y>n_z. A light ray incident perpendicular to the corneal surface (i.e., along the z-axis) experiences only n_x and n_y, which form a linear retarder with slow axis parallel to n_x and retardance equal to the product of n_x-n_y and corneal thickness. This tangential component of corneal birefringence is the one most commonly addressed in the literature. Because the cornea is curved, however, in most optical instruments not all incident rays are perpendicular to its surface and in general a ray experiences n_x, n_y and n_z. According to van Blokland and Verhelst [2], |n_x-n_z| is about 10 times larger than |n_x-n_y| and the retardance at oblique incidence should vary greatly with incident angle.

 

Fig. 1. Curved biaxial model of corneal birefringence. (A) A small slab of cornea showing the orientation of the coordinate system along the principal axes of the refractive index. Refractive indexes n_x and n_y are tangential and n_z is perpendicular to the corneal surface. (B) Cross section of a curved biaxial model through its apex. The section lies in the xz-plane and is illuminated by a beam of parallel rays. The red lines show the optical axes at three points. For two points on the model the refracted ray falls along an optical axis and experiences zero retardance (dashed lines). Note that the dimensions shown in this diagram do not correspond to those used to model the cornea (Sect. 3.3)

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The retarder formed in the general case is calculated from the intersection of an ellipsoid with semiaxes equal to n_x, n_y and n_z (the so-called refractive index ellipsoid or ellipsoid of wave normals) and the plane perpendicular to the incident ray that passes through the center of the ellipsoid [2,10]. This intersection is an ellipse with semiaxes equal to the effective refractive indexes experienced by the ray. A biaxial material has two incident directions for which this ellipse becomes a circle and along which a ray propagates with no change in polarization state. These directions are the optical axes of the material.

Figure 1(B) shows a beam of parallel rays incident on a curved slab, the geometry used by van Blokland and Verhelst [2], Götzinger, et al. [3], and in this study. The optical axes in the model, shown by red lines in Fig. 1(B), lie in the xz-plane at angles bisected by the z-axis. Thus, the curved biaxial model in a parallel beam exhibits two points with zero retardance where the refracted ray coincides with an optical axis, shown as two dashed lines in Fig. 1(B).

3. Methods

3.1 Subjects

The distribution of linear retardation over the pupil was measured in 21 eyes (10 right, 11 left) of 21 subjects (13 male, 8 female, ages 20–68). Eight eyes were measured on more than one occasion. Testing was performed in a darkened room to allow a subject’s eyes to dilate naturally. All subjects reported their corneas to be normal and none had undergone refractive surgery. No information was obtained on refractive error. The research study followed the tenets of the Declaration of Helsinki, was approved by the University of Miami Institutional Review Board and subjects provided written informed consent to participate.

3.2 Measurement of corneal birefringence

A scanning laser polarimeter (GDx-VCC^™, Carl Zeiss Meditec, Inc., Dublin, CA) was used without corneal compensation to obtain retardance images of the macular region of the eye [Fig. 2(A)]. These images typically show a “bow tie” pattern centered on the fovea, because the macula contains a birefringent structure, Henle’s fiber layer, composed of the radially oriented axons of the foveal photoreceptors [11]. The slow axis of retardation, which lies along the axons, is also radially oriented and the overall structure functions as an “intraocular polarimeter” to measure corneal birefringence; the line of maximum retardance corresponds to the slow axis of corneal birefringence and the modulation depth depends on corneal retardance [7,12]. Instrument software uses an algorithm that is insensitive to macular retardance to calculate the axis and single-pass retardance of the cornea from retardance values on a circle placed on the macular bow tie pattern. Corneal retardance is expressed in nm to give values that do not vary significantly with wavelength [13].

 

Fig. 2. Measurement of corneal birefringence with GDx-VCC^™ scanning laser polarimeter. (A) Macular retardance image with a bow tie pattern produced by the interaction of corneal birefringence with the radial birefringence of Henle’s fiber layer. The bow tie orientation coincides with the axes of corneal birefringence. Analysis of the retardance pattern around a circle centered on the bow tie provides the value of corneal retardance. (B) Acquisition screen with centered pupil. The yellow reticle fixed to the center of the image marks the center of the instrument scan pattern (ray O in Fig. 3) and specifies the horizontal and vertical axes in laboratory coordinates. (C) Acquisition screen with offset pupil. Data obtained in this configuration are plotted at the location of the center of the reticle relative to the pupil center (red dot).

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Prior to acquisition of a retardance image, the instrument images the anterior portion of the eye to enable focus and alignment, as shown in Figs. 2(B), 2(C). In clinical practice measurements are made as in Fig. 2(B) with the pupil centered in the scan. To map corneal birefringence in this study, the instrument objective lens was translated to move the scan pattern relative to the pupil, as shown in Fig. 2(C), and a transparent screen overlay with concentric circles was used to record the position of the scan center relative to the center of the pupil. The screen was calibrated by imaging a millimeter scale.

 

Fig. 3. Schematic of measurement optics. Left eye viewed from above. O: Central ray of scan pattern. F: Ray from the internal fixation target. Dashed blue lines: Translated objective lens and fixation ray.

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Because the corneal birefringence measurement was derived from the macular bow tie pattern, the corneal point sampled in a measurement was not the point located at the screen center, but rather was the point on the cornea intersected by the scan beam when it imaged the macula. The GDx-VCC^™ provides an eccentric, internal fixation target (created by briefly increasing the scan intensity) at a position selected to provide scan coverage of the peripapillary retina (Fig. 3). The scan beam was focused on the retina, the fixation target did not move with translation of the objective lens and for each corneal position the measurement beam (e.g., blue dashed line in Fig. 3) was parallel to ray F in Fig. 3. Thus the cornea was sampled with an array of parallel beams distributed over its surface.

3.3 Numerical model of curved biaxial cornea

A simple model of a curved biaxial slab (Fig. 1) was implemented and used to produce approximations to certain features of the measured data. To compare theory and data, it was necessary to position the model within the laboratory coordinate system with a spatial transformation that provided for translation or tilt of the model cornea relative to the eye, rotation of the eye to fixation and translation of the pupil relative to the scan center. For a ray incident at a point on the surface and parallel to ray F in Fig. 3, the refracted ray within the slab was calculated by Snell’s Law applied to the incident and normal vectors at the point. The principal refractive indexes determined a local coordinate system at each point, as shown in Fig. 1(A). The direction of the refracted ray in this local system and the equation of the refractive index ellipsoid (x ²/n ² _x+y ²/n ² _y+z ²/n ² _z=1) were used to calculate the birefringence experienced by the ray [14]. The resulting slow axis was transformed back into laboratory coordinates and the birefringence was multiplied by the length of the refracted ray within the slab to yield retardance.

Corneal parameters were taken from the LeGrand full theoretical eye [15]. The anterior and posterior surfaces of the cornea were spheres centered on the optic axis of the eye (not to be confused with the optical axes of birefringence) with radii of 7.8 mm and 6.5 mm, respectively. The corneal central thickness was 0.55 mm and its refractive index was n_y=1.3771. The direction for the corneal slow axis at each point (corresponding to n_x) was taken so that the projections of all local x-axes onto a plane perpendicular to the optic axis of the eye were parallel. All modeling and data analyses were implemented in MATLAB (The MathWorks, Natick, MA).

4. Results

The single-pass retardance of the central cornea of the 21 eyes ranged from 8 to 103 nm. Three patterns of corneal birefringence were observed and are described separately, but first a qualitative assessment of repeatability is presented.

4.1 Measurement repeatability

Measurements of central corneal retardation by GDx-VCC^™ are known to have low variability [6]. To assess the repeatability of the mapping technique used here, eight eyes were measured more than once. Because the measurement points were selected arbitrarily there is no one-to-one correspondence between data in the two sets (except at the central point 0,0) and the repeatability cannot be expressed in a simple statistically-derived number. However, visual inspection of data revealed that the corneal retardation, both retardance and slow axis direction, had the same pattern on different occasions, even months apart. Examples from two eyes (Fig. 4) show this repeatability and also show that a small region of very low retardance seen in the superior-temporal quadrant of both eyes fell within a 1 mm area on the two occasions (dashed circles in Fig. 4).

4.2 Pattern 1: Nasally downward corneal retardation

Figure 5 displays in two different formats an example of the corneal retardation pattern that, with variations, was found most frequently. The format in Fig. 5(A) shows the location of individual measurements and emphasizes the retardation axis, while the contour plot in Fig. 5(B) emphasizes the variation in retardance over the area measured. The central point (0,0) provides the corneal retardation that is compensated by the GDx-VCC^™ scanning laser polarimeter in clinical measurements of the RNFL [7,8].

 

Fig. 4. Repeatability of corneal retardation measurements. Each short line represents one retardation measurement. The line is centered on the measured point, is oriented in the direction of the slow axis and has length proportional to retardance. Black lines are the first data set and red lines the second. Data sets were aligned at the pupil center (0,0), then the central lines were displaced slightly from each other for clarity. A 1 mm diameter dashed circle encloses the region of minimum retardance in each eye. In all figures, N denotes nasal cornea, T denotes temporal cornea, the calibration bar above the T represents 100 nm single-pass retardance and an 8 mm diameter circle centered on the pupil is provided for reference. A) Right eye, measurement interval 11 months. Central retardation — black: 38 nm, 33.5° ND; red: 35 nm, 38.1° ND. B) Right eye, measurement interval 6 days. Central retardation — black: 33 nm, 23.3° ND; red: 36 nm, 22.8° ND.

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Fig. 5. Nasally downward retardation pattern. Measurements of single-pass corneal retardation in a left eye. (A) Retardation at each point is shown as in Fig. 4. Central retardation (red line at 0,0): 49 nm, 31° ND. (B) Contour lines (10 nm interval) show the variation of retardance with position. Gray fill: Elongated zone of approximately uniform retardance. In both panels the approximate locations of zero retardance (optical axes) are marked with red asterisks and the dotted line through the center is perpendicular to the central slow axis.

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The central slow axis in this eye, shown as a red line in Fig. 5(A), was 31° ND, within the 20°–40° range found in about 40% of normal eyes [1]. A line perpendicular to the central slow axis (dotted lines in Fig. 5), which we will call the pupil bisector, divided the corneal retardation pattern into approximately symmetric halves. Slow axes measured near the pupil bisector were approximately parallel to the central slow axis. Two small regions located on opposite sides of the pupil bisector at 3.3 and 2.7 mm from the center had very low retardance (red asterisks; minimum measured value=1 nm). We interpreted these as regions where the refracted incident rays fell along optical axes of the cornea, as depicted schematically in Fig. 1(B). Far from the center the slow axes were oriented approximately circumferentially, as exemplified best between 2 o’clock and 8 o’clock in Fig. 5(A). The overall retardance distribution in this eye was saddle-shaped, as seen in Fig. 5(B), with the seat of the saddle forming an elongated zone (gray fill) of approximately uniform retardance and axis located near and inferotemporally to the pupil center.

Figure 6 shows measurements from three more corneas with a nasally downward pattern. The corneal data in Fig. 6(A) show all of the features of Pattern 1: symmetry around a line perpendicular to the central slow axis, a saddle-shaped retardance distribution with an elongated zone of uniform retardation, parallel slow axes near the pupil bisector, circumferential slow axes near the pupil margin and two optical axes. The examples in Fig. 6(B), 6(C), however, do not show the symmetry seen in the previous two examples. They have the features of Pattern 1 in the superotemporal pupil and have a similar elongated zone of uniform retardance that crosses the midline, but in the inferonasal pupil no second optical axis was found and in the direction where an optical axis was expected [approximately 3:30 and 4:00 o’clock, respectively, in Figs. 6(B), 6(C)] the peripheral slow axes did not become circumferential within the pupil.

 

Fig. 6. Three more corneas with a nasally downward pattern, shown in the same format as Fig. 5. (A) Symmetric pattern. Left eye; data combined from three measurement sessions. Average central retardation=35 nm, 40° ND. (B) Asymmetric pattern. Right eye, same data as red lines in Fig. 4(A). Central retardation=35 nm, 35° ND. (C) Asymmetric pattern. Right eye. Central retardation=40 nm, 38° ND.

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4.3 Pattern 2: Horizontal corneal retardation

Three eyes had a striking pattern in which the slow axis of retardation was nearly horizontal over most of the measured pupil. Two of these corneas had the two highest values for central retardance (62 and 103 nm) and all had nearly horizontal central slow axes. Figure 7 shows one example. Retardance decreased to a very low value at the temporal edge of the pupil, suggesting the location of an optical axis (asterisk).

 

Fig. 7. Horizontal retardation pattern. Right eye with high central retardance (62 nm) oriented nearly horizontally (5° ND).

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4.4 Pattern 3: Low central retardance

We purposely recruited two subjects from a previous study of corneal birefringence [1] who had very low central retardance. Figure 8 shows an eye that had one of the lowest central retardances previously measured (13 nm), shown as the small red line at 0,0 in Fig. 8(A). The contour map for this eye, shown in Fig. 8(B), had a broad area of low retardance, highlighted here by the 20 nm contour (dashed red line). The precise location of an optical axis was poorly defined; here one was placed between the two lowest measurements. Outside the low retardance area, retardance increased rapidly in the supero-temporal and -nasal directions, but more gradually inferiorally. As in previous figures, the dotted lines are oriented perpendicular to the central slow axis, although symmetry similar to that seen in other eyes was not apparent.

 

Fig. 8. Low retardance pattern. Right eye, central retardation: 13 nm, 21° ND. The dashed red line in B shows the 20 nm contour surrounding a large area of low retardance.

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4.5 Summary

We measured the distribution of corneal retardation in 21 eyes of 21 patients with a wide range of central corneal retardance. The measurements were deemed repeatable by qualitative assessment of multiple measurements in eight eyes. Three retardation patterns were observed. Pattern 1, nasally downward retardation, could be characterized as either approximately symmetric around a pupil bisector perpendicular to the central slow axis [Figs. 5 and 6(A)] or asymmetric [Figs. 6(B), 6(C)]. Pattern 1 was seen in 16 eyes, of which five were approximately symmetric, eight were asymmetric and three were uncertain due to small pupil size. Pattern 2, horizontal retardation over most of the measured pupil and high central retardance, was seen in three eyes. Pattern 3, a large area of very low retardance, was seen in two eyes. Note that this distribution of patterns does not represent the population distribution, both because the sample size is small and because subjects with low central retardance were specifically recruited based on results of an earlier study [1].

5. Model

To test the hypothesis that corneal birefringence behaves like a biaxial material, we used a simple model (Sect. 3.3) to calculate retardation patterns similar to those displayed in the previous section. Parameters in the model - n_x, n_z (n_x≥n_y>n_z) and the slow axis of central retardation — were adjusted to approximate the measured pattern. Special attention was given to superimposing at least one optical axis of the model onto observed areas of minimum retardance (presumed optical axes) and to approximating the retardance gradients around the minima. Small translations of the corneal model relative to the optic axis of the eye were made to align the patterns, if necessary. For clarity, points with calculated retardance values greater than 120 nm were not displayed.

5.1 Pattern 1: Nasally downward corneal retardation

 

Fig. 9. Curved biaxial model of cornea with parameters chosen to approximate the data in Fig. 5. The black dashed line connects the optical axes of the model. (A) Individual points. Red lines are the model calculated on a 0.5 mm grid, blue lines are data from Fig. 5(A). The black dotted line is the pupil bisector drawn perpendicular to the central slow axis of the data. (B) Retardance contours. Model contours (red, 20 nm intervals) superimposed on the data contours of Fig. 5(B) (blue, 10 nm intervals).

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Figure 9 compares the uniform biaxial model to the data of Fig. 5. The model, calculated with a central axis of 27° ND, birefringence values of n_x-n_y=1.0×10^-4 and n_y−n_z=11.5×10^-4, and translations of -1.1 mm horizontally and +0.2 mm vertically, shared several features of the data. Of foremost importance, the two optical axes of the model were easily matched to the two retardance minima of the data, suggesting that these minima occur where rays refracted by the curved cornea coincide with optical axes of the material, as illustrated schematically in Fig. 1(B). Further, Fig. 9(A) shows that the directions of the slow axes for model and data generally agreed over the corneal surface, with parallel slow axes near the pupil bisector and circumferential slow axes near the pupil margin. The model also had a saddle-shaped retardance distribution with an elongated area of uniform retardation, as seen in Fig. 9(B), but unlike the data the saddle point fell near a straight line between the two optical axes (dashed line), rather than in a zone displaced infero-temporally. Thus, the model contours were more symmetric around the line joining the optical axes than the data contours. No adjustment of model parameters, including translation and tilt of the model surfaces relative to the eye, could displace the saddle point of the model toward the observed position.

5.2 Pattern 2: Horizontal corneal retardation

Figure 10 compares the uniform biaxial model to the data of Fig. 7. The model was calculated with a central axis of 7° ND, birefringence values of n_x-n_y=1.2×10^-4 and n_y-n_z=9×10^-4, and a translation of +1.0 mm horizontally and -0.3 mm vertically. The biaxial model described the data reasonably well when the nasal optical axis was allowed to fall outside the pupil margin. As in Pattern 1, the slow axis directions of the model and data were similar but, unlike Pattern 1, the saddle point of Pattern 2 data fell much closer to the line connecting the optical axes of the model (dashed line in Fig. 10).

 

Fig. 10. Model (red) with parameters chosen to approximate the data in Fig. 7 (blue) in the same format as Fig. 9. (A) Individual points. (B) Retardance contours.

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5.3 Pattern 3: Low central retardance

Pattern 3 may represent a qualitatively different distribution of corneal birefringence than Patterns 1 and 2. The uniform biaxial model could not reproduce the large area of low retardance with nearby rapid retardance rise. On the assumption that this cornea did not have uniform birefringence, we ignored the nasal side of the pupil and chose equal x and y values for refractive index, that is, a uniaxial model, to compare to the data. We selected parameters to produce a 20 nm contour of about the same spatial extent and offset as in Fig. 8(B).

 

Fig. 11. Uniaxial model with parameters chosen to approximate the low retardance area in Fig. 8. The pupil bisector (black dotted line) emphasizes the asymmetry of this birefringence pattern. (A) Measured (blue) and calculated (red) retardance and axis. (B) Retardance contours: Model (red) superimposed on data (blue). The dashed red line shows the 20 nm contour caused by a 25% decrease in birefringence.

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Figure 11 compares this uniaxial model to the data of Fig. 8. The model was calculated with birefringence values of n_x-n_y=0 and n_y-n_z=21×10^-4, and a translation of -0.35 mm horizontally and +0.15 mm vertically. The single optical axis of the uniaxial model was perpendicular to the surface and was located near the center of approximately circular contours. To provide a sense of the sensitivity of the uniaxial model to birefringence change, the dashed red line in Fig. 11(B) shows the 20 nm contour for n_y-n_z=16×10^-4 ; a 25% decrease in birefringence produced a 15% increase in the diameter of the contour.

In the temporal half pupil the model and data are quite similar in both slow axis and retardance. In the nasal half, however, the axes of the uniaxial model circled around its center whereas the data axes streamed off toward the nasal pupil margin in a way that suggests the biaxial pattern of Fig. 10.

6. Discussion

We measured corneal retardation over the surface of the cornea with a GDx-VCC^™ scanning laser polarimeter. This instrument uses the radial birefringence of Henle’s fiber layer as an intraocular polarimeter to measure the slow axis and retardance experienced by light that passes from a fixation point to the fovea of the eye. Although in clinical practice the instrument is aligned on the center of the pupil, we found that repeatable measurements could be made with it aligned on any point in the dilated pupil (Fig. 4). The result was a set of corneal measurements obtained with an array of parallel incident beams, an illumination geometry similar to those of van Blokland and Verhelst [2] and Götzinger, et al. [3]. Our study was designed to test the curved biaxial crystal model of corneal birefringence proposed by van Blokland and Verhelst [2].

6.1 Observations

We observed three general patterns of corneal retardation. In the most common pattern (Pattern 1, Figs. 4–6), the slow axes of the central measurement and of all measurements near a line perpendicular to the central slow axis sloped nasally downward (ND). Retardance contours surrounded an optical axis (point of zero retardance) in the superotemporal pupil; then, as retardance increased toward the pupil center, the contours opened up into an elongated region of relatively uniform retardance that also sloped ND. In some examples of Pattern 1, a second optical axis was located in the inferonasal pupil [Figs. 5, 6(A)] and in others the pattern of slow axes and retardance extended to the inferonasal pupil margin without encountering a second optical axis [Figs. 4, 6(B), 6(C)]. In another pattern (Pattern 2, Fig. 7), seen occasionally, the slow axes were nearly horizontal across the entire cornea, with an optical axis apparent near the temporal margin of the pupil. Pattern 2 was associated with higher central retardance, consistent with the correlation between retardance and axis found in one earlier study [1], but not in another [6]. Rarely (Pattern 3, Fig. 8), in cases purposely sought out for having low central retardance, there was a large area of low retardance just superotemporal to the pupil center.

An absolute limitation on the SLP measurement method is the need for macular imaging, i.e., a measurement ray must pass through the pupil of the eye. Variation in pupil size accounts for the variation in measured corneal area among the examples in Figs. 4–8. Even with pharmacological dilation, however, it would be unlikely that SLP measurements could be made of the corneal limbus, of interest because birefringence there may be dominated by circumferential collagen fibrils [16].

6.2 Modeling

A simple model of a curved biaxial material with uniform properties showed reasonable agreement with several features of Patterns 1 and 2, but also important differences. The optical axes of the model matched the observed retardance minima and mimicked the pattern of slow axis directions over large regions of the cornea [Figs. 9(A) and 10(A)]. Model retardance contours, however, were more symmetric around a line joining the optical axes than data contours, which in Pattern 1 showed an inferotemporal displacement of the saddle point. These differences between model and data suggest that the model’s assumption of uniformity is not correct, but rather that corneal birefringence varies with location. Also, as is well known, the peripheral cornea deviates from a sphere, but this alone cannot account for the differences between model and data. For these reasons, the values for n_x and n_z used to calculate the models in Figs. 9–11 should be regarded as only approximations to the actual birefringence.

The spatial variation of corneal birefringence was especially apparent in Pattern 3 (Figs. 8, 11). The large area of low retardance could be modeled by setting the tangential refractive indexes n_x and n_y equal (the model becomes uniaxial with its optical axis perpendicular to the corneal surface), but this approach failed completely to capture the inferonasal retardation, which resembled the ND patterns. We hypothesize that n_x gradually increased relative to n_y, i.e., the cornea became biaxial, on a path between temporal and nasal pupil.

6.3 Evidence for biaxial corneal birefringence

The evidence appears strong that at any given point (perhaps with the exception of some areas in Pattern 3) the cornea behaves as a curved biaxial material with its fastest principal axis perpendicular to the corneal surface and its slowest axis usually oriented ND, the model proposed by van Blokland and Verhelst [2]. The evidence is as follows.

1) When measured with rays from a single direction, many corneas display two small areas with low or zero retardance [e.g., Figs. 5 and 6(A)]. These retardance minima appear where a ray refracted by the curved cornea coincides with an optical axis of the material [Fig. 1(B)]. In other corneas that display only one such area, the possibility of a second, and thus a second optical axis, can be inferred from the retardation pattern [e.g., Figs. 4(B), 6(B), 6(C) and 7]. Two retardance minima have also been shown qualitatively in corneal images made with circularly polarized light [17].

2) The pattern of slow axis directions can be mimicked over large regions of the cornea by a simple curved biaxial model [e.g., Figs. 9(A) and 10(A)].

Note: Points 1 and 2 above are consistent with the data in Ref. 2 on which van Blokland and Verhelst based their model.

3) Measurements of central cornea with the converging rays of a petrological microscope show hyperbolic isogyres characteristic of a biaxial material with its fast axis perpendicular to the surface and slow axis oriented ND [5].

4) The corneal stroma is composed of 200 or more thin birefringent layers with their optical axes parallel to the corneal surface (along the direction of collagen fibrils) and with the optical axes of adjacent layers lying at large angles to each other. Theoretical analysis of a stack of two or three such birefringent layers shows that, in general, the stack behaves as a biaxial material, with two optical axes oriented obliquely to its surface [4]. In the special case of two layers oriented with their respective optical axes perpendicular to one another (e.g., the model proposed in Ref. 3), the stack is uniaxial with its optical axis perpendicular to its surface.

In an apparent contradiction to biaxiality, Götzinger, et al. [3], did not find two optical axes, but rather a uniaxial pattern centered on the point of perpendicular incidence. Their method, however, may not have been sensitive enough to detect biaxiality. Retardance noise estimated from their en face color maps may be on the order of 25–50 nm. This noise would obscure the small retardances shown in this study and thus the difference between n_x and n_y, leaving n_z as the major contributor to their PS-OCT images. The variation in n_z direction with corneal position would produce their observed radially-increasing retardance and circumferentially-oriented slow axes. At any particular point, the direction of n_z would not be expected to change much with depth, also consistent with their observations. Thus, the data in Ref. 3 are not sufficient to reject the biaxial model.

6.4 Consequences of biaxial cornea for scanning laser polarimetry

Assessment of retinal nerve fiber layer (RNFL) birefringence with SLP is an important optical tool for glaucoma diagnosis and management [9]. Accurate RNFL measurements require a means to compensate for corneal birefringence. The laser scan pattern that forms a retinal image necessarily enters the cornea over a large area. Due to the biaxial nature of most corneas, rays that enter at oblique angles experience birefringence from both tangential (n_x,n_y) and perpendicular (n_z) corneal axes. Corneal compensation as currently performed for SLP (e.g., GDx-VCC^™) only neutralizes the tangential component [7,8]. As shown in Fig. 12, because the eye fixates nasally to center the optic disc, the beam scanning the nasal retina (N) enters the cornea more obliquely than the beam scanning the temporal retina (T), thus interacting more strongly with n_z to add a corneal contribution (R) to retinal retardance maps. The effect of a biaxial cornea on the diagnostically important RNFL regions may be small, but cannot be dismissed without further study.

 

Fig. 12. Schematic demonstrating in a left eye the possible consequences of a biaxial cornea for SLP. This GDx-VCC^™ image of a patient with severe glaucoma exhibits weak RNFL birefringence. (N) Beam directed toward nasal retina. (T) Beam directed toward temporal retina. (M) Uniform macula shows good compensation of the tangential component of corneal birefringence in temporal retina. (R) Retardance signal on nasal retina due to biaxial cornea.

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7. Conclusions

Three conclusions can be drawn from the data and modeling presented here. First, a complete description of corneal birefringence requires that the corneal stroma be treated as a biaxial material. Useful simplification can occur with perpendicular incidence but must be extrapolated with caution (e.g., Fig. 12). Second, the three patterns illustrated here and the wide range of central corneal retardance found in much larger samples [1,6] show that corneal properties vary greatly among people. Third, birefringence within a single cornea varies significantly with position. Further characterization of corneal birefringence should address all of these points.

The ideal method for complete characterization of corneal birefringence may not yet exist. Quantitative measurements will require multiple incident angles at many points on the cornea. Measurement sensitivity must be comparable to SLP (1–2 nm of retardance) to detect small values es of n_x-n_y. The method should be capable of measuring birefringence at the corneal limbus, where collagen is organized differently than in the center [16]. To capture the full range of central retardance values, the measurements must be obtained from a large number of people, so the method should work in vivo and be relatively rapid. Future studies of corneal birefringence await the development of such a method.