Fan Zhou, Xin Hong, Donald T. Miller, Larry N. Thibos and Arthur Bradley
| A corneal aberrometer based on SH wavefront sensing was developed and validated using calibrated aspheric surfaces. The aberrometer was found to accurately measure corneal reflective aberrations from which corneal topography and corneal refractive aberrations were derived. Measurements of reflective aberrations correlated well with theory (R2 = 0.964 to 0.994). The sag error RMS was small, ranging from 0.1 to 0.17 μm for four of the five calibrated surfaces with the fifth at 0.36 μm due to residual defocus. Measured refractive aberrations matched with theory and whole-eye aberrometry to within a small fraction of a wavelength. Measurements on three human corneas revealed very large refractive astigmatism (0.65-1.2 μm) and appreciable levels of trefoil (0.08-0.47 μm), coma (0.14-0.19 μm), and spherical aberration (0.18-0.25 μm). The mean values of these aberrations were significantly larger than the RMS in repeated measurements |
Monochromatic aberrations of the eye degrade retinal image quality and lead to poor visual performance 1-8. Defocus and astigmatism (clinical sphere and cylinder) are the leading culprits 8, 9, but higher-order aberrations such as coma and spherical aberration are also present in normal, pathologic and surgically-altered eyes 5, 8, 10, 11. Clinical aberrometers – based on Shack-Hartmann (SH) wavefront sensing technology – have recently become available to quickly, noninvasively and objectively measure monochromatic aberrations of the whole eye 4, 6, 9, 11-24. Whole eye aberrometry provides a powerful means to predict the optical performance of the eye and ultimately the quality of the retinal image 25-30.
Optical characteristics of specific ocular structures (e.g. cornea and crystalline lens) and individual optical surfaces, however, are not revealed by whole eye aberrometry. Such information is fundamental to understanding optical normalcy of individual ocular structures and surfaces; the optical interaction and possible balancing of aberrations between the various ocular surfaces 31, 32; and the improved design of contact and intra-ocular lenses.
Placido rings have been used in recent years for measuring the aberration content of the anterior corneal surface of the eye 10, 15, 33-41. This technology is very different from SH wavefront sensing. However, inferring aberrations of the cornea from conventional corneal topography data is fairly straightforward. For example, deviations in topography are directly proportional to wavefront deviations with a constant of proportionality equal to the difference between the air and corneal refractive index. Although computer assisted photokeratoscopy provided improved accuracy and resolution when introduced in 1980, there are reasons to question its ability to accurately measure corneal aberrations at the same sub-wavelength level 42-45 achieved by whole eye aberrometers. Also, because of the radially symmetric ring targets used in modern topographers, topographic distortions tangent to these rings cannot be measured (the skew ray problem) 46-50.
A potential method for overcoming the limitations of Placido disk topographers is to re-engineer the SH aberrometer as a corneal topographer. This approach has several key advantages over existing corneal topographers that are based on Placido ring technology. A SH aberrometer can be easily aligned to the subject’s line of sight, the recommended reference axis for predicting foveal image quality 51. Current topographers on the other hand, align to the videokeratoscopic (VK) axis 52-54, which cannot be directly used to determine foveal image quality. A SH aberrometer is immune to the skew-ray error because both tangential and saggital slopes of the cornea are measured. Furthermore, a SH instrument can sample the cornea at high density enabling small surface anomalies to be resolved. In this paper, for example, we employ a sampling density of 7 to 8 lenslets/mm compared to 4 rings/mm for a conventional topographer. Lastly, only a small modification is required to convert a whole-eye SH aberrometer into a SH corneal topographer. Thus it should be possible to combine both instruments into a single device, with resulting savings in cost, space, and user training.
In this paper, we introduce the concept of a SH instrument that measures the reflective aberrations of the cornea from which corneal topography and refractive aberrations are obtained. We validate the methodology using calibrated model eyes in conjunction with whole-eye aberrometry and ray tracing models; and we show initial results and analysis of the topography and wave aberration content of the anterior corneal surface in three human eyes using the SH corneal aberrometer. Early accounts of this work were presented at the 2001 and 2003 ARVO meetings 55, 56.
Figure 1 shows the apparatus of the combined SH whole-eye and corneal aberrometer. Strictly speaking, the Shack-Hartmann sensor (consisting of the micro-lenslet array, the CCD video camera, and the computer which analyzes the data images) is a sub-system of the aberrometer. The rest of the aberrometer is designed to perform three functions: (1) form a point source of laser light on the retina (whole-eye aberrometer) or curvature center of the anterior cornea (corneal aberrometer) that will be a source of reflected light captured by the CCD camera, (2) provide a fixation target via lens L3, and (3) focus the eye’s pupil in the plane of the lenslet array via lenses L4 and L5 (and also L7 for the corneal aberrometer) so that the wavefront aberrations are analyzed in the pupil plane.
The whole-eye aberrometer is essentially a fundus camera which takes multiple pictures of a single spot of light reflected from the retina. The retinal spot is produced by a narrow (~1 mm) laser beam (λ=633μm) focused onto the retina by the eye’s optical system. Multiple images of this spot are captured by a custom fundus camera fitted with a final imaging lens consisting of an array of small lenslets placed one focal length from a video sensor. This micro-lenslet array partitions the reflected wavefront of light emerging from the eye into a large number of smaller wavefronts, each of which is focused to a small spot on the sensor. Assuming the wavefront is approximately linear over each individual lenslet aperture, and that the lenslet has no odd-symmetric aberrations, then the spatial displacement of the centroid of the resulting image relative to the optical axis of the lenslet is a direct measure of the local slope of the incident wavefront. Integration of the array of slope measurements reconstructs the phase of each point on the wavefront relative to other points on the same wavefront. By this method, a single brief exposure of the sensor, plus subsequent computer processing of the captured image, yields a detailed picture of the overall shape of the reflected wavefront and the eye which produced it.
The micro-lenslet array (Adaptive Optics Associates, Boston, MA) in our system is a square array of lenslets with center-to-center spacing = 0.4 mm, which provides about 100 measurements of the wavefront shape over a 6.2 mm pupil. Using the computational method described by Liang et al. 2, we fit the original slope data to the derivatives of Zernike circle polynomials by the method of least squares. These regression coefficients are used to represent the aberrated wavefront as a weighted sum of Zernike basis functions. We routinely compute a vector of 66 Zernike coefficients (i.e. orders 0-10) with the SH aberrometer. All computations are performed in MATLAB (Mathworks, Inc.). We used the right-hand coordinate reference frame and the double-index convention for naming the Zernike coefficients and polynomials recommended by the OSA/VSIA Standards Taskforce51. In this convention the Zernike coefficient is defined as the standard deviation of the wavefront attributable to each Zernike polynomial in units of microns. When compared with the results of ray tracing models, which were restricted to polynomials up to 7th order, the SH data was truncated to the same order. Additional details of the whole-eye aberrometer can be found elsewhere4.
In contrast to the whole eye aberrometer, isolation of the anterior corneal aberrations was realized by capturing light reflecting from the outer corneal surface (Purkinje I). This reflected wavefront is corrupted by only the aberrations at the anterior cornea and pre-corneal tear film.
Two key optical/mechanical hardware modifications are required to convert the SH whole eye aberrometer into the SH corneal aberrometer (Fig. 1). First, a high numerical aperture and high quality (i.e. low monochromatic aberration) objective lens (L7) is strategically inserted in front of the subject’s eye. The lens images the curvature center of the cornea onto the SH CCD array and a plane tangent to the cornea itself is conjugated with the lenslet array. Second, in order to obtain reflections from a large area of the cornea, the diameter of the beam entering the objective lens is expanded. This is realized by the addition of a telescopic system (L6 and L8). As a secondary change for alignment purposes, a channel was added to monitor the point spread function formed by the reflected wavefront at the cornea (PB3, video camera).
Axial translation of the high NA lens L7 permits incident beam vergence at the cornea to be set equal to corneal curvature. In other words, the focal point of objective lens L7 is positioned on the nominal curvature center of the cornea, C. This allows incident light to retro-reflect and produce an approximately collimated beam that is sampled by the lenslet array of the SH sensor, which provides about 1,850 measurements of the wavefront shape over a 6.2 mm diameter region of the cornea. The resulting focused spots are recorded with a CCD camera from which corneal reflective aberrations are calculated. Corneal topography and refractive aberrations are computed from the corneal reflective aberrations and measurements of the nominal corneal radius (see below). Although not implemented here, the high sampling resolution of the lenslet array could be used with Zonal reconstruction to measure fine surface features not captured by Zernike modes (e.g. tear film topography).
Because the objective lens focal point is coincident with the center of curvature of the cornea (Fig. 2), the measured corneal diameter in the apex plane is proportional to the lens’ NA and the radius of the eye’s cornea (eqn.1). The actual diameter DC, which is employed in this paper, is about 90% of the apical diameter DA due to curvature of the cornea.
(Measured Cornea Dia at Apex) = (Cornea Radius) * (2*NA) (1)
The wavefront sampled by the lenslet array is corrupted by the aberrations of the corneal front surface (pre-corneal tear film), but also those indigenous to the measuring instrument. The instrument was designed to contribute low levels of aberrations by using high quality stock doublets for all lenses, except L7. Because L7 must have a large numerical aperture in order to measure a large corneal area, stock lenses of this NA would have noticeable monochromatic aberrations that would contaminate the reflective corneal signature.
A custom objective lens (L7) was manufactured to provide diffraction-limited performance across a ±1.5 degree field of view (subtended at the nodal point of L7) at the illumination wavelength of 0.633 μm; a NA of 0.5; and effective and back focal lengths of 36 and 29 mm, respectively. This field of view is sufficient to allow lateral displacement of the eye up to ± 0.2 mm without loss of image quality. A 0.5 NA in conjunction with a 36 mm focal length allows measurement of the central 6.2 mm diameter (DC) area of the typical cornea, while providing a reasonable lens diameter of 36 mm. A 29 mm back focal length provides a sufficient working distance from the subject’s eye. A custom lens was designed and manufactured to these specifications by J. L. Wood Optical Systems. Performance of the lens was confirmed with a commercial interferometer.
As an additional step for improved system performance, a separate measurement using a calibrated sphere (which has zero reflective aberrations) in place of the cornea was routinely subtracted from each corneal measurement. In principle this removes any fixed instrument aberrations that may be present. In fact, the RMS of the instrument aberrations was measured at 0.26 ± 0.012 μm over a seven month period. On a relative scale, this is exceedingly small contributing about 0.06% of the mean reflective aberrations measured in three human eyes in this study.
If measuring a spherical cornea, the SH corneal aberrometer depicted in Figure 1 generates a perfectly regular array of point images at the CCD. Therefore, the aberrometer provides wavefront slope measurements relative to a sphere in contrast to whole-eye aberrometers that measure wavefront deviations from a plane. This difference is the basis for an additional stage of algorithmic processing required beyond that for whole-eye aberrometry. Corneal aberrometry requires transformation of reflective wavefront slope measurements to actual corneal slope, then to corneal topography and finally refractive wave aberrations. The components of this reconstruction process and our implementation of it are detailed below.
The surface profile of the cornea can be reconstructed from the reflected rays shown in Fig. 3A. The back focal point of the objective coincides with an average center of curvature of the actual cornea (Caverage). Because the actual cornea lies inside the back focal point (OO’ < F), the objective forms a virtual image B’ of any arbitrary point B on the actual cornea. The objective lens converts a perfectly spherical cornea (B) into a perfectly flat virtual cornea (B’). Slope deviations (θ) from this flat (virtual) surface are measured with the SH topographer by conjugating this apparent surface to the lenslet array.
For a spherical cornea, a ray traveling along AB will retro-reflect back onto BA causing θ and Φ to both equal zero. In this simplified case, the corneal slope at B (relative to that at the apical point O’) is equal to α, the angle of the incident ray relative to the optical axis of the system. The ray pattern shown in Fig. 3A depicts a reflection from a typical corneal shape that is aspheric with flattening in the periphery. The ray reflects at angle Φat point B and exits the objective at angle θ that is now non-zero. The corneal slope at B is α - Φ/2. Thus at any arbitrary point B on the actual cornea, α represents the slope profile of a perfectly spherical cornea and Φ/2 represents deviations from this spherical shape. As illustrated by the block diagram at the bottom of Fig. 3A, our reconstruction model (1.) transforms measured slope values (θ) to actual slopes (α - Φ/2) and (2.) converts (α - Φ/2) to topography and then finally to refractive aberrations.
We developed an iterative model to transform measurements of θ to α - Φ/2. The model starts with an initial spherical estimate of BC from which α - Φ/2, then corneal topography, and finally a second estimate of BC is computed. α - Φ/2 is calculated from measurements of BC, α, and θ. The new value of BC is fed back into the model for the next iteration. The model assumes the distance AB’ is independent of small height deviations in the apparent surface specified at B’. The model was found to converge rapidly in two to three iterations with a termination criteria of less than 10-6 μm RMS change in the estimated corneal topography between successive iterations. Accuracy of this iterative solution will be discussed in the result section.
Actual slopes (α - Φ/2) are converted into reflective aberrations expressed as a Zernike expansion using the same least squares fitting software employed in the SH whole eye aberrometer. The topographical profile of the cornea surface was rendered by summing the reflective aberrations and measured corneal radius. Refractive aberrations of the model cornea were then derived by ray tracing through the rendered corneal surface as described in Fig. 3B.
For convenience, Figs. 3A and 3B depict the incident and exiting rays in the same meridian. Our reconstruction model is more general than this because we numerically computed the propagation of rays in three dimensions. Therefore our method was fully capable of tracing all rays generated by radially asymmetric corneas including skew rays.
The SH corneal aberrometer was assessed by measuring reflective aberrations from calibrated radial symmetric and then asymmetric model eyes. Measurements on the asymmetric model included skew rays. Measurements on the symmetric model were further assessed by transforming reflective aberrations and measured nominal radius of curvature into corneal topography (sag) and then to corneal refractive aberrations.
Six ellipsoidal surfaces were employed as calibrated model corneas to assess instrument accuracy. Each was lathed onto the end of a transparent PMMA (polymethymethacrylate) dowel having a refractive index of 1.49 (λ=633μm). The physical diameter and height of the dowels were 11 and 19.5 mm, respectively. The focal point of the ellipsoid was just outside the physical dowel behind the opposite surface, which was itself optically flat. The model eyes were manufactured by Sterling International Technologies, which guarantees the ellipsoidal surface elevations to ±1.0 μm of the specified value across the central 10 mm diameter zone. Samples of the surfaces were verified by the manufacturer using Rank Taylor Hobson Talysurf, a device that makes stylus measurements to a resolution better than 0.1 μm. The manufacturer claims that these surfaces are therefore accurate to 0.1 μm, though they guarantee 1.0 μm accuracy. The ellipsoidal profiles are described by Baker’s formula y2= 2rz – pz2, where r and p are the apical radius and shape factor of the surface, respectively. Values of r and p for the six surfaces are given in Table 1. Surfaces were chosen that represent the range of most normal corneas with the 78-07 approximating the average 54.
| Surface # | Apical radius (r) | Shape factor (p) | Description |
| 78-10 | 7.8 mm | 1.0 | reference sphere with average corneal radius |
| 78-07 | 7.8 mm | 0.7 | average (prolate) cornea |
| 78-05 | 7.8 mm | 0.5 | average radius, acute prolate cornea |
| 78-13 | 7.8 mm | 1.3 | average radius, oblate cornea |
| 73-07 | 7.3 mm | 0.7 | small radius, average prolate cornea |
| 83-07 | 8.3 mm | 0.7 | large radius, average prolate cornea |
The average human cornea has an r and p value near 7.8 mm and 0.7, respectively.
As a precursor to laboratory measurements on the model eyes, accuracy of the reconstruction algorithm to convert measured SH slope values (θ) into corneal topography was quantified in terms of sag error. Sag error is defined as the difference in sag between the reconstructed and theoretical surface profiles. The latter is derived from the r and p values in Table 1. To avoid measurement errors that would bias the performance of the algorithms, theoretical values of θ were generated using commercial ray tracing software (Zemax-SE Optical Design Program, Version 10.0) that modeled the ellipsoidal corneas and objective lens L7. A detailed blueprint of the custom lens was made available by Wood Optical Systems and incorporated into the ray tracing model.
For the laboratory measurements, each of the six ellipsoidal surfaces were mounted in turn on an XYZ translation stage and its optical axis aligned to that of the SH instrument in Fig. 1.
Transverse alignment in X and Y followed a two step protocol. First, reasonable alignment was achieved by visually monitoring the array of raw SH focal spots on the computer monitor, which were updated several times per second. Vignetting was purposely designed into the topographer to occlude focal spots when the cornea was transversely misaligned, and this provided a visually apparent indicator for the operator. The system was designed to vignette for corneal offsets greater than ±0.1 to 0.2 mm (depending on corneal shape) assuring that misalignment greater than this would result in unusable SH data. Second, more precise alignment was achieved by minimizing vertical and horizontal coma (Z31 and Z3-1), which were measured with the SH instrument. Coma was empirically found to be sensitive to small misalignments.
Each model cornea was axially aligned to the confocal and “cat’s eye” positions as illustrated in Fig. 4. The axial distance between the two points represents the corneal radius averaged across the illuminated patch of cornea. In the confocal position, the corneal curvature center is coincident with the focal point of the objective (L7 in Fig. 1). In the cat’s eye position, the focal point is located at the corneal apex. Axial alignment followed a two step process similar to that performed for X and Y alignment. Rough alignment to the confocal and cat’s eye positions was achieved by using a video camera to visually optimize the PSF formed by the reflected wavefront as the model cornea was axially translated. More precise alignment followed by minimizing defocus (Z20) in the wavefront measurements collected with the SH instrument in both configurations.
Reflective corneal aberrations were measured with the focal point of lens L7 at the confocal position. In addition the cornea was conjugated to the lenslet array of the SH sensor by translating in tandem L7 and the model cornea. The aberration measurements included residual system aberrations of the SH instrument. These unwanted anomalies were largely removed by subtracting a separate measurement of a calibrated sphere (r = 7.8 mm, p = 1.0).
In addition to the measurements of the radially symmetric model corneas positioned on the aberrometer axis, the ellipsoidal corneas were deliberately rotated horizontally about their apex so that the optical axis of the instrument no longer coincided with the axis of the model eye. Due to the asphericity of the ellipsoid surfaces, a small translation of the cornea, i.e. transverse to the SH optical axis, was necessary to re-direct the reflected light from the rotated models back onto the lenslet array. Translation was measured with a micrometer. A rotation of 3.8 degrees was found sufficient to induce noticeable amounts of asymmetric aberrations. Results were obtained for a rotation of 3.8 deg (plus one additional measurement at 7.6 deg) and a translation typically between 0.5 and 0.7 mm. Accuracy was assessed by comparing measurements of reflective aberrations (astigmatism, coma, trefoil, spherical aberration, total aberrations) to theoretical predictions obtained by modeling the aspheric corneas.
The impact of corneal surface shape on retinal image quality and ultimately vision is dependent on the surface’s ability to refract light. We took advantage of the fact that the model corneas have only one refractive surface. Therefore the refractive aberrations of corneas calculated from reflective aberrations should match the "whole-eye" aberration measurements for these corneas. As a further validation experiment, SH corneal aberrometry measurements for the six symmetric surfaces were converted into maps of refractive wave aberrations using a refractive index of 1.49 for the PMMA material (see Figure 3B). These were compared to theoretical predictions obtained with ray tracing (Zemax) and refractive wave aberrations measured with the SH whole-eye aberrometer (Fig 1) for the same model eyes. A difference in RMS between measured and theoretical aberration maps below λ/14 — a standard definition for diffraction-limited image quality — indicates measurement errors that negligibly impact the quality of the retinal image relative to that of the theoretical surface.
Whole-eye measurements were realized with an artificial retina positioned at the back focal plane of the ellipsoidal surface. It turned out that because the back optically flat surface of the PMMA dowel was slightly in front of the retina, wavefronts propagating to and from the retina must pass through the PMMA-air interface. Theoretical ray tracing confirmed that the back surface typically increased the total RMS of the system by 0.026 μm and represented about 4.5% to 20% of the total RMS for all model eyes except for 78-05. For this eye the back surface was 49% of the total RMS, but this is because the total RMS was very small (0.048 μm). Consequently for a meaningful comparison of the refractive aberrations derived from corneal topography with those from the "whole eye" aberrometry, the topography-based data were modified to include the predicted contribution of the back PMMA surface. Both sets of experimental results were compared to the ray tracing predictions which included the back surface. Refractive aberrations across the central 6.2 mm of the cornea (astigmatism, coma, trefoil, spherical aberration, and total aberration) for the three methods (whole-eye aberrometry, corneal aberrometry, and theoretical ray tracing) were compared.
Measurements using the SH corneal aberrometer were collected from the right eyes of three subjects with healthy corneas(aged 36, 28, and 32). Spectacle sphere and cylinder, which was obtained by a professional subjective refraction, was –1.75/-1.5x88, +5D, and –3.75/0.75x178 for subjects’ DM, JW, and JQ, respectively. None of the subjects wore contact lenses habitually. A dental impression mounted on an XYZ translation stage provided accurate positioning of the eye. The subject’s line of sight was aligned to the optical axis of the SH instrument using a fixation target and a video camera to monitor the location of the subject’s pupil. The average radius of the anterior cornea was measured by finding the cat’s eye and confocal positions. Both were located by visually optimizing the PSF formed by the corneal reflection and minimizing defocus (Z20) in the wavefront measurements collected with the SH instrument. Three interleaved measurements were collected and averaged for the two positions. The physical separation of the average cat’s eye and confocal positions was taken as the average corneal radius across the measured area of the cornea.
Reflective aberrations of the cornea were measured by positioning the subject’s eye at the confocal position and conjugating the subject’s cornea with the SH lenslet array. The latter was realized by translating in tandem the objective lens and eye. For subjects JW and JQ, SH measurements were collected in three consecutive sessions, each consisting of nine measurements captured in less than 4 seconds. A total of 27 measurements (three sessions) were collected within 30 minutes. The subjects were taken off the bitebar and realigned between sessions. For subject DM, measurements were collected for only one session.
A 10 msec exposure time was used for all measurements. For each subject, the raw CCD images were evaluated for signs of vignetting and other eye motion artifacts that would prevent processing. The useable images were averaged and reflection aberrations were computed. System aberrations were effectively removed by subtraction of the measured aberrations of a calibrated reference sphere (78-10). Reflective aberrations were combined with measurements of average corneal radius to obtain corneal topography and then refractive wave aberration maps. The refractive index of the human cornea was specified at 1.376.
Exposure level was 2.6 microwatts, more than 110 times below the maximum permissible exposure recommended by the American National Standards Institute for continuous illumination. This represents the most conservative case in which the incident beam is inadvertently focused onto the retina during the alignment procedure57.
Accuracy of the reconstruction algorithm for converting corneal radius and reflective aberrations to corneal topography is shown in Fig. 5. Differences between the topography calculated from the predicted reflective aberrations and those calculated from the algebraic description of the ellipsoidal surfaces define the sag error of the surface reconstruction algorithm. The radially-averaged sag errors for symmetric and asymmetric corneas are plotted as a function of radial position relative to the corneal center. For the symmetric case (Fig. 5A), sag error is essentially zero for all models up to a radius of 1.5 mm (central 3 mm of cornea) at which point it begins to increase. For the four prolate and oblate surfaces the maximum error stays below 0.18 μm and 0.25 μm, respectively. The very close overlay of the four prolate surfaces (which encompass a wide range of human corneal shapes) suggests a systematic error that if subtracted could improve the accuracy of the algorithm. Horizontal rotation and translation (see methods) had little impact on sag error measured in the prolate models (Fig. 5B), but increased the sag error in the oblate model cornea from -0.25 μm to -0.41 μm.
Fig. 6 shows the correlation between measured and predicted reflective aberrations for the five calibrated model corneas. Correlations are given for the Zernike coefficients Z22 (horizontal astigmatism), Z31 (horizontal coma), Z40 (spherical aberration), and total aberrations (astigmatism plus 3rd through 7th Zernike orders). Each scatter plot in Figure 6 contains five data points representing the symmetric corneas plus nine points for the same corneas horizontally rotated by ±3.8 deg and laterally translated between 0.5 and 0.7 mm. The plots contain an additional point for the 78-05 model that was rotated by 7.6 deg and translated by 1.1 mm.
As shown in Fig. 6, measurement and theory were highly correlated in all four scatter-plots with a linear regression fit showing R2 values ranging from 0.964 to 0.994.
For the five symmetric corneas, astigmatism, coma, and trefoil were close to zero as expected. Spherical aberration on the other hand was noticeably large for all corneal shapes with measured and predicted values ranging from 1 to 2 μm RMS. Unlike the case of refracting surfaces, reflective spherical aberration was positive for the four prolate corneas due to peripheral flattening and negative for the single oblate surface due to peripheral steepening.
For the asymmetric corneas, astigmatism and coma were the most sensitive to rotation. For example rotation of the acute prolate cornea (78-05) from 0 to 7.6 degs increased the measured astigmatism and coma from essentially zero to 2.79 and -4.36 μm RMS, respectively. In general, an increase in rotation (with all other parameters fixed) resulted in an increase in astigmatism and coma regardless of the r and p surface profile. A change in the direction of rotation (e.g. +3.8 to –3.8 deg) resulted in a sign reversal for coma and trefoil. Although not shown, their complementary 3rd order polynomials Z3-1 (vertical coma) and Z3-3 (vertical trefoil) were, as predicted, insensitive to both the sign and magnitude of the horizontal rotation. Astigmatism and spherical aberration were also insensitive to rotation.
Although the observed and predicted aberrations shown in Figure 6 are highly correlated, in some cases the regression slopes and intercepts deviated significantly, but slightly from the Y=X line, e.g.Y=0.927X for Z22, and Y=1.115X for Z40 indicating small but systematic errors in the data. These systematic errors were related to outliers in these regression plots. Interestingly, the outliers in the data originated from the model corneas least like the typical human cornea. Specifically, the 78-05 (most prolate) model that was rotated 7.6 deg and translated 1.1mm generated the top right most point in the astigmatism plot. Also, the 78-13 (most oblate) model with 0 and 3.8 degree rotation and 0, 0.5, and 0.7mm translations generated the 3 points in the bottom left of the spherical aberration plot. Without these outliers, all four regression analyses produced slopes very close to 1.0 (mean 0.984, range 0.948 to 1.026) and intercepts very close to zero (mean 0.003, range 0.01 to –0.06) and none of these differences between the model predictions and the experimental data were statistically significant. Thus, for a restricted range of prolate corneas, this analysis confirms that the SH corneal aberrometer is able to generate accurate estimates of reflective astigmatism, coma and spherical aberration. By including the maximally rotated highly prolate cornea and the significantly oblate cornea, small but statistically significant errors appear in the data.
Accuracy of the SH instrument for reconstructing the surface topography was assessed using the five symmetric model corneas. In this case, surface topography reconstructed from measured reflective aberrations was compared to the topography calculated algebraically for each surface. Differences between these are defined as experimental sag errors. Fig 7A shows the radially-averaged sag error for the central 6.2 mm diameter of the cornea and Fig 7B shows the corresponding 2D sag error maps.
Comparison of the measured sag error in Fig. 7 to that generated solely by the reconstruction algorithm in Fig. 5 reveals that the use of measured values for the corneal radius and reflective aberrations (rather than theoretical ones) increases the overall sag error above that produced by the reconstruction algorithm alone. Up to a radius of 1.5 mm, the sag error is near zero for three of the models with the remaining two monotonically increasing to about 0.2 μm. Above 1.5 mm two of the models maintain a maximum error no greater than 0.2 μm. The most prolate cornea (p=0.5) has a maximum error of –1 μm; the smallest cornea (r=7.3 mm) has a maximum error of 0.48 μm; and the single oblate cornea reaches a value of 0.4 μm.
A representative raw CCD image collected on the 78-05 surface is shown in the upper left of Fig. 7B. The image illustrates the pristine appearance of the ~1,850 focal spots that densely sample the cornea. The highly defined appearance of individual focus spots contrasts that routinely observed in SH whole-eye raw images where significant scattering in the retina and internal ocular media noticeably broadens the spots. Also shown in the same figure are 2D contour maps of sag error. Measurement errors in the position of the confocal and cat’s eye (which are used to determine the average corneal radius) manifest themselves as defocus in the sag error maps. This is strikingly evident in the contour map for the most prolate cornea (p=0.5) where defocus is the dominant error. This surface was found the most difficult to axially align, and we suspect its large amount of spherical aberration produced a large depth of focus that resulted in a higher positioning error. The RMS sag error across four of the surfaces was small, ranging from 0.1 to 0.17 μm. The RMS of the 78-05 surface was much larger (0.36 μm) due to the presence of significant defocus.
As a final validation step, topographic maps of the five symmetric model corneas were converted into refractive aberration maps and compared to theory as well as whole-eye SH measurements obtained on the same corneas. Due to the radially symmetric shape of the five corneas, refractive astigmatism, coma, and trefoil are expected and were observed to be approximately zero.
Theoretical and measured results for spherical aberrations and total aberrations are shown in Fig. 8. For a spherically converging wavefront, the ideal reflector shape that imparts no aberrations onto the wavefront is a sphere. In refraction however, it is well known that this same surface produces substantial amounts of positive spherical aberration. The perfect refractive shape that mitigates spherical aberration for imaging a distant target is an ellipse with a shape factor (p) of 0.55, which is calculated from 1-1/n2, where n is set to 1.376 for the cornea.58 Larger and smaller shape factors lead to a monotonic and rapid increase in spherical aberration. This relationship is observed in the results of Fig. 8A. The predicted refractive Z40 values of the three 7.8 mm radius corneas are –0.025, 0.17, and 0.83 μm RMS for p = 0.5 (the far left lower point), 0.7 (the middle point in the central three points), and 1.3 (the far right upper point), respectively. The acute prolate model (78-05) generated the least refractive spherical aberration and the oblate (78-13) the most. Therefore, although the acute prolate provided the best refractive image quality, its reflective aberrations are the highest. Due to the predominance of spherical aberration in these surfaces, the total aberrations (Fig. 8B) mirror those seen for Z40.
As shown in Figure 8 the predicted and observed aberrations using the SH corneal aberrometer and whole-eye aberrometer are highly correlated. The linear regression fitting R2 values for Z40 and total aberrations are 0.975 and 0.969 respectively. The regression analyses also produced slopes very close to 1.0 (mean 0.961, range 0.835 to 1.087 for Z40 and mean 0.911, range 0.778 to 1.044 for total aberrations) and intercepts very close to zero (mean -0.011, range -0.061 to 0.0391 for Z40 and mean 0.03, range -0.023 to 0.083 for total aberrations). None of these differences between the model predictions and the experimental data were statistically significant. Thus, this analysis confirms that the SH corneal aberrometer is able to generate accurate estimates of refractive spherical and total aberrations.
To assess how measurement errors of the instrument might affect retinal image quality, the difference in refractive RMS between theory and SH corneal aberrometry were compared to the diffraction threshold of 0.045 μm. This threshold corresponds to λ/14 for the 0.633 μm aberrometer wavelength. Differences in the refractive RMS were 0.045 μm (78-07), 0.05 μm (78-05), 0.075 μm (78-13), 0.036 μm (73-07), and 0.059 μm (83-07). Note that defocus (Z20) was removed prior to calculating the RMS. These results reflect differences in image quality relative to that of the corresponding theoretical surface. Four of the five RMS values are below or very close to the diffraction threshold. The remaining cornea, which is oblate (p = 1.3), is about λ/8 for the 0.633 μm aberrometer wavelength.
Corneal radius and reflective aberrations were measured on three human corneas using the SH corneal aberrometer. Of the 9 SH measurements collected on DM and the 27 collected on JQ and JW, vignetting and other eye motion artifacts – which precluded some of the focal spots from being captured - reduced the useable sample size to 3, 9, and 18, respectively.
The surface profile of the cornea was reconstructed from these measurements, from which ellipsoidal surfaces described by Baker’s formula were fitted by minimizing the RMS sag difference. Fitted r values were 7.85, 7.69, and 8.04 mm and fitted p values were 0.85, 0.81, and 0.67 for DM, JW, and JQ, respectively. Of the five model corneas, 78-07 was the closest match to the three subjects and will be used below for comparison.
RMS values for several reflective Zernike coefficients plus the total aberrations are shown in Fig. 9. Astigmatism was the dominate aberration for all three eyes with RMS values ranging from 2.52 to 5.13 μm. For DM trefoil was the second most significant aberration at 2.01 μm; for JW it was coma (1.05 μm); and for JQ it was spherical aberration (1.08 μm). The total aberrations was strongly shaped by the high levels of astigmatism with the least overall influence arising in subject DM due to his relatively large amount of trefoil.
Comparison to the reflective aberrations for the 78-07 model cornea (3.8 deg rotation) reveals that astigmatism and trefoil are noticeably higher in the three human eyes, while coma and spherical aberration are at comparable levels. Astigmatism in the human eyes is five to ten times higher than the maximum observed for the 78-07 cornea even with a 3.8 deg rotation. Coma levels in the human corneas and the 78-07 model straddle 1 mm RMS. Trefoil varies from 0.2 to 2 μm in the human eyes, and is essentially zero in the model cornea even with rotation. Spherical aberration varies the least among the three subjects with a value near 1 μm. This is comparable to the reflective spherical aberration in the 78-07 cornea as is expected given the similar p values. Total aberrations in the human eyes are 1.8 to 4.5 times larger than in the 78-07 cornea.
The corresponding 2D refractive wavefront maps are shown in Fig. 10 for the three human corneas. Decomposition of the maps into RMS values for several Zernike coefficients plus total aberrations are shown in Fig. 11. The figures reveal that astigmatism – like that for the reflective aberrations - is the dominate anomaly in all three subjects with RMS values ranging from 0.60 to 1.41 μm. It is interesting to note that although DM has -1.5 D of spectacle cylinder, the amount measured at his cornea was only 0.6 D (0.65 μm RMS as shown in Fig. 12), which suggests that internal optics, most likely the crystalline lens, is the contributing source. JQ on the other hand had a smaller amount of spectacle cylinder (0.75 D), yet corneal measurements indicate 1.4 D (1.2 μm RMS). This suggests a balancing of cylinder between the anterior cornea and internal optics in this particular eye. For DM trefoil is the second most significant aberration at 0.5 μm, with its characteristic three prong shape visibly apparent in the contour map (Fig. 11). For the other two subjects, spherical aberration is second largest although significantly less than astigmatism. Averaged across the three subjects, coma has the smallest magnitude. Because rotation of elliptical model corneas generates significant levels of coma (see Figure 6), the small amount of measured coma in these three human corneas suggests that the optical axis of the cornea is closely aligned with the primary line of sight. Note that the refractive aberrations in Fig. 11 are roughly four to five times smaller than their reflective counterparts in Fig. 9, although there is no theoretical rule that we know of that should govern this relationship.
In relation to the refractive results obtained on the 78-07 model cornea, the human eyes were significantly more aberrated both in terms of individual Zernikes as well as total aberrations, the latter of which was at least 4.5 times larger in the human eyes owing to the large amount of astigmatism these eyes shared. The lone exception was spherical aberration where both human and model were about 0.2 μm RMS. This agreement should be expected given the strong dependence of spherical aberration on r and p values, which in this comparison are reasonably close. The relatively large amount of trefoil in two of the subject’s eyes reflects a significant limitation of corneal modeling with ellipsoidal surfaces which lack any trefoil even when rotated.
As a precursor to laboratory measurements, the accuracy of the reconstruction algorithm was examined for both symmetric and asymmetric corneas. The maximum sag error induced by the algorithm for the four prolate surfaces and single oblate surface was found to be no worse than 0.22 and 0.41 μm, respectively. Height deviations in the corneal surface advance or retard the local refracting wavefront by Δz(nc-1), where Δz is the elevation difference and nc the refractive index of the cornea (1.376). Maximum sag errors of 0.22 and 0.42 μm correspond, therefore, to refractive wavefront deviations of 0.08 (~λ/8) and 0.16 μm (~λ/4), respectively, for the HeNe wavelength (λ=0.633 μm) employed by the SH aberrometer. From a large population study of wavefront aberrations in the normal human eye, the RMS across a 6 mm pupil for 3rd through 7th Zernike orders is, on average, about 0.3 μm. The corresponding maximum error (assuming four times the RMS value) is 1.2 μm. Assuming the cornea contributes a large fraction of this then a reasonable goal for instrument accuracy (i.e. sag error) would be five times smaller or 0.24 μm. In this context the errors of the reconstruction algorithm are at least a factor of one and half smaller than this goal.
The ability of the SH corneal aberrometer to accurately detect corneal reflective aberrations was further assessed by comparing RMS measurements to theoretical predictions on five calibrated model eyes. RMS was compared for individual 3rd and 4th order Zernike coefficients as well as their collective impact with coefficient values covering a broad range: astigmatism (-0.5 to 3 μm), coma (-4 to 3 μm), spherical aberration (-1 to 2 μm), and total aberrations (0 to 6 μm). Linear regression fits showed very high correlations with R2 values ranging from 0.977 to 0.994. Interestingly, two of the corneal profiles (78-13 and 78-05) were consistently more difficult to measure than the others. Both are furthest (of the five tested) from that of the average human cornea. For the 78-05, we suspect axial misalignment with the SH instrument – which was made difficult by the cornea’s high asphericity – played a large role in this error. For the 78-13, we do not have any tangible evidence at this point to indicate where the source of the error may be at. For model corneas more closely approximating the human cornea we found that the SH corneal aberrometer was highly accurate (observed = predicted aberrations).
The corneal topography was reconstructed from measurements of the radius and reflective aberrations for symmetric model corneas. Reconstructions were also attempted on asymmetric corneas, but the relatively large rotation (3.8 deg) and translation (0.5 to 0.7 mm) used in this study made the theoretical corneal sag predictions highly sensitive to small measurement errors in the rotation and translation. These measurement errors – effecting only the theoretical model and not an indication of the SH instrument accuracy – were found sufficiently large to mask errors incurred by the instrument itself. Nevertheless the generalization of reflective aberrations to corneal topography is captured by the use of symmetric data as any performance difference between symmetric and asymmetric corneas would manifest itself in the reflective aberrations, which was clearly not found in this study. Also, from a theoretical perspective, there is no fundamental reason why the SH instrument should perform differently for symmetric and asymmetric corneas.
Maximum sag error across all symmetric corneal models was no greater than 1 μm. This is a very conservative estimate of the instrument performance as the 1 μm is almost surely due to residual defocus stemming from axial misalignment by the operator. Nevertheless, this corresponds to a refractive wavefront deviation of 0.376 μm, which approaches the desired instrument accuracy of 0.24 μm. If we exclude the defocus term, the instrument accuracy easily exceeds this standard (e.g. the maximum sag error for the 73-07 model was 0.48 μm) For comparison, typical commercial corneal topographers have a maximum error ranging from 2 to 16 mm with an average value around 6 μm for a 6 mm pupil37 .The SH instrument provides an order of magnitude improvement over this average value reflecting the high sensitivity of this instrument.
Refractive aberrations were derived from the reconstructed corneal topography. Differences in the experimental and theoretical refractive RMS ranged from 0.036 to 0.075 μm for the five model corneas. Four of the five RMS values are below or close to the diffraction threshold with the remaining one (which is oblate) less than λ/8.. These suggest that the SH instrument has sufficient sensitivity to detect subtle surface anomalies that induce near-diffraction-limited perturbations in the transmitted wavefront. The dynamic range of the instrument is such that these subtle anomalies are detectable in the presence of large irregularities that generate large aberrations (e.g. the highly aspheric 78-05 cornea).
It is interesting to note that the SH corneal aberrometer on average produced measurements that were closer to theory than the SH whole-eye aberrometer, an instrument that now is widely considered to be the gold standard for whole-eye wave aberrations. The corneal aberrometer provided a better measurement for three of the five model surfaces and an equally close measurement for a fourth. Both the whole-eye and corneal aberrometers had difficulty with the acute prolate (78-05).
SH measurements were obtained on three subjects from which reflective and refractive aberrations of the anterior cornea were determined. The reflective aberrations were dominated by astigmatism in all three eyes with coma, trefoil, and spherical aberration present at appreciable levels. Large variations in astigmatism, coma and trefoil were found between subjects. The corresponding refractive aberrations were also dominated by astigmatism with trefoil and spherical aberration a distant second, and coma the least significant. Comparisons were made with the calibrated model cornea 78-07 that possessed r and p values similar that of the subjects tested. The results clearly revealed major differences between the two with the human eyes being substantially more aberrated and containing a wider range of Zernike aberrations. These early results on human eyes suggest that conic surfaces – a common tool for evaluating and calibrating corneal topographers 44 poorly represents the cornea. Conic surfaces appear to be a good model for corneal spherical aberration, but not for radially asymmetric corneas.
Because the cornea is highly curved, its reflection is sensitive to eye motion. A pressing concern for the corneal aberrometer then is whether the human eye can be sufficiently stabilized to permit meaningful corneal measurements. Our results suggest that a bite bar stage used in conjunction with controlled instrument vignetting are sufficient to produce repeatable measurements as indicated in Figure 11. The standard deviation across repeated measurements was found small compared to the mean. Furthermore, coma and spherical aberration had similar standard deviations (see Figure 11) even though coma is highly sensitive to lateral motion of the eye and spherical aberration is not. This suggests that the eye was sufficiently stable that motion error was not the limiting variable. Although instrument vignetting occluded the focal spots in many of the measurements (reducing the useable sample size to 33% to 66% of that collected), from a practical standpoint this low yield could be easily offset by collecting measurements at a high rate and automatically discarding those deemed unusable. In addition, it is likely that a larger percentage of useable images would accrue from improved instrument design and protocol. In fact we have observed an increase with operator experience.
In this study we employed the SH instrument as a corneal aberrometer by capturing light reflecting from the precorneal tear film (Purkinje I). By the same line of reasoning, essentially the same design could in principle measure wavefronts that generate Purkinje images II, III, and IV. Using the equivalent mirror theorem 59 in conjunction with knowledge of refractive indices and physical separations of the ocular media would allow us to isolate the optical performance of individual surfaces such as those of the natural crystalline lens or a surgically inserted intraocular lens.
We have developed and validated a new type of corneal aberrometer based on SH wavefront sensing. Validation relied on measuring the reflective aberrations of calibrated aspheric surfaces and from which corneal topography and refractive aberrations were derived. The aberrometer was found to accurately measure corneal reflective aberrations; generate submicron resolution topographic maps; and create refractive aberration maps that match with theory and whole-eye measurements to within a small fraction of a wavelength. Measurements were obtained on three subjects and revealed appreciable levels of astigmatism, trefoil, coma, and spherical aberration with mean values significantly larger than the RMS in repeated measurements
Financial support was provided by National Eye Institute grant R01-EY05109 to L. N. Thibos. This work has been supported in part by the National Science Foundation Science and Technology Center for Adaptive Optics, managed by the University of California at Santa Cruz under cooperative agreement No. AST-0043302.
Figure 1. Combined Shack-Hartmann whole-eye and corneal aberrometer. Lenses L6, L7, and L8 are inserted for measuring corneal topography and aberrations (see text for details). Other channels monitor the point spread function (PB3, video camera) and pupil position (PB4, video camera), and present a fixation target (PB1, L3).
Figure 2. Objective lens (L7) with a high numerical aperture allows aberration measurements over a large cornea section.
Figure 3. (A) The schematic diagram demonstrates the geometry of the cornea + objective system, as well as the incident and the reflected rays. See text for the algorithm used to reconstruct the corneal surface profile from the measured reflective wave aberrations. (B) Refractive aberrations of the model cornea derived from corneal topography. The refractive aberrations were calculated based on the optical path difference between the marginal path (ABC) and the paraxial path (DOC) from distant object to paraxial image.
Figure 4. Determining the apical radius of the cornea requires finding the (a) cat’s eye and (b) confocal positions. At both positions, the objective lens creates a reflective wavefront with nominally zero defocus.
Figure 5. (A) Radially-averaged sag errors generated by the reconstruction algorithm for five symmetric model corneas positioned on the aberrometer axis. The algorithm converted theoretical SH slopes (θ) into surface topography. (B) Radially-averaged sag errors for the same five model corneas with the surfaces rotated (3.8 deg) and laterally translated (0.5, 0.6, and 0.7 mm) to induce skew rays. Sag error is relative to the theoretical surface shape of each model cornea.
Figure 6. Correlation between measured and predicted reflective aberrations for Z22, Z31, Z40, and total aberrations. The latter includes astigmatism plus 3rd through 7th Zernike orders. The plots contain all measurements obtained on the five examined model corneas for both symmetric and asymmetric orientations.
Figure 7. (A) Radially-averaged corneal sag error for the SH corneal topographer. SH slope measurements (θ) were obtained for each of the five model corneas. Error is relative to the theoretical surface shape of the model cornea. (B) Upper left is a representative raw CCD image that illustrates the very high sampling rate of the SH corneal topographer. Also shown are 2D contour maps of measured sag error for the SH corneal aberrometer. Contours cover the central 6.2 mm of each of the five cornea models examined. RMS sag error is displayed in the lower right of each plot. Contour lines represent 0.2 μm change in surface elevation.
Figure 8. Correlation between measured and predicted (A) refractive Z40 and (B) total aberrations in the whole eye for the central 6.2 mm of the cornea. The latter includes astigmatism plus 3rd through 7th Zernike orders. The plots contain independent refractive measurements on five model eyes with the SH whole-eye (triangles) and corneal (circles) aberrometers (Fig. 1). Predictions were based on a ray tracing model that incorporated the theoretical corneal shape.
Figure 9. Reflective wave aberrations across the central 6.2 mm of the cornea for three healthy subjects. Average RMS values are plotted for Z22 and Z2-2 (astigmatism), Z31 and Z3-1 (coma), Z33 and Z3-3 (trefoil), Z40 (spherical aberration), and total aberrations (astigmatism plus 3rd through 7th Zernike orders). Measurements were obtained with the SH corneal topographer.
Figure 10. Refractive wave aberrations measured across the central 6.2 mm of the cornea for three subjects. Contour lines represent 0.32 μm change relative to a planar (perfect) wavefront. Measurements were obtained with the SH corneal aberrometer.
Figure 11. Refractive wave aberrations across the central 6.2 mm of the cornea for three healthy subjects. Average total RMS values are plotted for Z22 and Z2-2 (astigmatism), Z31 and Z3-1 (coma), Z33 and Z3-3 (trefoil), Z40 (spherical aberration), and total aberrations (astigmatism plus 3rd through 7th Zernike orders). Measurements were obtained with the SH corneal topographer. Error bars represent one standard deviation of the measurements.
