Ocular wavefront tomography (OWT) is a computational process for customizing a wide angle schematic eye to achieve the twin goals of anatomical accuracy and functional equivalence.21 Anatomical accuracy is achieved by constraining the parameters of the model to lie within acceptable limits. Functional equivalence is achieved by adjusting the model's parameters until the wavefront aberration function of the model along the foveal line-of-sight, and along multiple peripheral lines-of-sight, match the aberrations measured in an individual eye or some representative eye. The OWT procedure consists of four steps. Step 1 configures a generic model eye as an initial template that serves as a starting point for the optimization process. This initial model should include all of the anatomical features deemed important. Depending on the intended application, this template model eye might include a single refracting surface, or multiple surfaces, or a gradient index lens. Step 2 determines the wavefront aberrations of the eye along multiple lines-of-sight that adequately sample the range of visual field to be corrected. For example, a modified clinical Shack-Hartmann aberrometer22,23 or a scanning Shack-Hartmann Aberrometer24 could be used to obtain such aberration measurements in an individual eye. Alternatively, the eye might be characterized by aberrations of a typical eye across the visual field for a target population. Step 3 formulates a good measure of the dual customization goals (anatomical similarity and functional equivalence) in the form of a merit function (eqn. (1)) that quantifies the degree to which the current state of the model satisfies the design objective. The first part of the merit function represents the anatomical similarity between the customized model and the anatomical dimensions common to all eyes or, if available, the specific dimensions measured for an individual eye. The second part of the merit function measures the difference between the wavefront measurements of individual eyes along multiple lines-of-sight and the theoretical values obtained by ray tracing through the customized model. The relative weighting of these two parts, and of the various factors within each part, is flexible and application-dependent. 21 Step 4 formulates the tomography problem of adjusting the template model eye to become anatomically similar and functionally equivalent to the subject's eye into an optimization problem of finding a customized model eye that achieves a global minimum of the merit function. A variety of optimization techniques can be used for this purpose, including damped-least squares, simulated annealing, neural networks, case-based reasoning, and expert-system. These computational-intensive techniques solve the optimization problem iteratively.

Given a wide-angle model of an eye, the OWT technique can be used to optimize the design of a contact lens used in conjunction with the eye. We do this by fixing the parameters of the eye model while optimizing the parameters of the contact lens to achieve the desired optical behavior of the eye + lens system across the visual field. Again the method involves four steps: the construction of a design template, the specification of design goals, the formulation of a merit function, and the optimization of the design. Steps 3 and 4 are the same as described above in section 2.1, but the first two steps require modification as described below.

To design a contact lens using OWT, the first step is to create a design template consisting of a generic contact lens in apposition to a fixed model eye. The parameters of this generic contact lens will be iteratively optimized while the fixed model eye remains unchanged as a master system in the template. The choice of model eye depends on the specific application. For example, to customize a contact lens for an individual eye, the eye model could be obtained from wavefront aberration measurements by the OWT technique, using the corneal topography data in the merit function to ensure accurate geometry of the corneal front surface. Alternatively to design a more generic contact lens for improving peripheral vision for a population of eyes, a statistical model of the eye based upon population data may be preferred. In the examples reported below, we used a published model eye20 to enable a comparison with known results from the literature.

The second step of OWT lens design is to specify the desired optical performances of the lens + eye system in central and peripheral visual fields. In this study, we developed two different design goals. Both goals aimed to optimize image quality (e.g. RMS of wavefront error) along the central line-of-sight, but they differed for the peripheral field. Test case 1 aimed to improve image quality in the periphery whereas case 2 aimed to manipulate the variation of peripheral refractive error across the visual field for the purpose of myopia control.

Design goals for wide field-of-view require defining the optical performance (e.g. wavefront error) along peripheral lines-of-sight, where oblique viewing of the iris causes the entrance pupil to appear elliptical. Several methods are available to define wavefront aberrations over elliptical pupils, which we compare and contrast elsewhere.25,26 Particularly for test case 2, in order to calculate the spherical refractive error along the oblique line-of-sight it was convenient to use the scaling method22,26 which stretches the elliptical pupil into a circle of constant diameter for all lines-of-sight. Zernike aberration coefficients obtained from the stretched wavefront map can be converted to spherical refractive error using the formula derived by Atchison et al (eqn. B24 in Atchison's paper27). By ignoring the higher order terms in the original equation, the formula becomes

With the starting template set up in step 1 and rigorously formulated design goals in step 2, the merit function can be readily formatted in step 3. Similarly to the classic OWT approach, the first part of the merit function was formulated to measure the difference between the specified design goals and the ray tracing prediction of the design template (contact lens + model eye). The second part of merit function incorporates the mechanical constraints (e.g. edge thickness) of the contact lens from the fabrication or peripheral zone design. This merit function (Eq. (3)) is analogous to the merit function that measures the functional equivalence and anatomical similarity (Eq. (1)) of model eyes in section 2.1. The weighting inside and between each part of the merit function are flexible and can be iteratively adjusted during the design stage to achieve the balance of the design.28–30 After formulating the merit function, the optimization engine can be applied to find the optimal design that achieves the global minimum of the merit function in the final step.

A rotationally symmetric, wide-angle model-eye20 was chosen as the fixed master system in the OWT design template. This widely cited model captures the main anatomical features of the human eye with minimum complexity. Besides a spherically curved retina, this model eye consists of four refractive surfaces: anterior & posterior cornea and anterior & posterior lens. The model exhibits realistic off-axis aberration performance at moderate field degrees (10°–45°).20,22 To simulate axial myopia (−2D, 550 μm), the length of the schematic eye was increased appropriately (posterior chamber length = 17.1005 mm). Entrance pupil diameter was set at 5 mm, which is large enough to include significant amounts of higher-order aberrations.

Test case 1 was a monofocal aspheric lens optimized to improve peripheral optical quality (RMS of wavefront error). The template contact lens used rigid material with a refractive index of 1.492. The back surface of the contact lens was spherical with the same radius of curvature as the anterior corneal surface. A thin tear film layer of refractive index 1.336 was placed between the anterior cornea and the posterior surface of the contact lens. The front surface of the contact lens for test case 1 was aspheric with a surface sag z given in Eq. (4),

Test case 2 was a two-zone bifocal designed to correct the central vision and an outer annular zone designed to introduce relative myopic refraction pattern in the peripheral field. Its surface had sag profiles given in Eq. (5)

The goal of test case 1 was to correct foveal wavefront aberrations while simultaneously improving peripheral image quality of the Navarro myopic eye (−2D) Since foveal vision correction is of high priority, 80 % of the weight was assigned to minimize the RMS of central wavefront. The remaining 20 % of the weight was equally assigned to reduce the RMS of the wavefronts along other oblique line-of-sight within 50 degree visual field. To prepare the starting design template, we fit the myopic model eye with an aspheric design (CV in table 1) that solely gives the diffraction-limited performance for the central vision. After OWT optimization, we found a design (PV in table 1) that corrects the central vision and meanwhile improves the peripheral image quality.

Figure 1 (a) shows the root-mean-square (RMS) off-axis wavefront error of lens + eye as a function of field angle of the myopic Navarro model eye. Design CV is a traditional design that only aims to correct refractive error in central vision (CV). This design provided diffraction-limited performance centrally and served as the starting template for optimization with OWT. Design CV also improved image quality in the near periphery (field angle < 30°) but image quality in the far periphery was actually worse with the lens than without for reasons described later. By comparison, the design PV gives priority to correcting foveal refractive error while simultaneously aiming to improve image quality for peripheral vision (PV). This design reduced RMS wavefront error along the foveal line-of-sight to 0.14 μm (1/4 wave) over a 5 mm pupil and improved peripheral image quality out to 45° visual field relative to the uncorrected eye. Design PV provided superior image quality compared to design CV for all peripheral field angles. The slight penalty for this improved performance in the periphery was a small residual wavefront aberration (1/4 wave RMS) along the central line-of-sight (compared to the design CV's diffraction limited central correction).

Figure 1.

Performance of customized contact lens that improves the peripheral optical quality (5 mm on-axis entrance pupil, 550 nm). (a) Root-mean-square (RMS) of wavefront errors (lower- and higher- order aberrations) as a function of field angle of peripheral lines-of-sight for Navarro myopic eye (empty triangle), the design CV (solid circle), and the design PV (solid diamond); (b) The spherical refractive errors along lines-of-sight for Navarro myopic eye (empty triangle), the design CV (solid circle), and the design PV (solid diamond).

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The primary effect of the contact lens is to change mean spherical refractive error M as computed by eqn. (2). As shown in Figure 1b, M is slightly larger for design CV than for design PV at the fovea, but M increases more rapidly with field angle for design CV than for design PV. The relatively small amount of refractive error in the periphery for design PV is the primary factor that accounts for the superior image quality of this design relative to the other two conditions shown in Figure 1a. Other factors (oblique astigmatism and higher-order aberrations) play a smaller, but significant, role also. Peripheral astigmatism becomes larger for design PV than design CV. The astigmatism of PV and CV at 50° field angle are 4.49 μm and 3.65 μm respectively. On the other hand, the coma term of design PV is about 25 % smaller than design CV. The net effect of these aberration change is superior peripheral optical quality for design PV compared to design CV as shown in Figure 1a.

The goal of test case 2 was to correct foveal wavefront error while simultaneously changing relative peripheral refractive errors from hyperopic (in the Navarro model) to myopic (in the corrected eye). This optimized design (Design BF) is a concentric, two-zone bifocal design (Eq. 5) based upon the 5 mm on-axis entrance pupil of myopic model eye. Similar to test case 1, a merit function was set up with 60 % of the weight assigned to minimize the RMS of central wavefront error and the remaining 40 % of the weight was equally distributed over the peripheral lines of sight to manipulate the peripheral refractive pattern. The starting design template for this test case is summarized in the first row of Table 2. After OWT optimization, the bifocal (BF) design in Table 2 achieves the dual goals of correcting the central refractive error and changing the peripheral refractive pattern from relative hyperopia to relative myopia.

The BF design achieved diffraction-limited performance (5 mm on-axis entrance pupil) along the foveal line-of-sight by using the inner zone to correct the myopic eye's on-axis wavefront error with an asphericity that produces negative spherical aberration to compensate for the eye's positive spherical aberration. The sign of peripheral refractive error of design BF remains hyperopic in the near-peripheral visual field (< 35°), but beyond 35 degree visual field, the sign changes to myopic. This pattern of peripheral refractive errors is markedly different from the CV design, for which peripheral refractive errors are always hyperopic. This result reveals the design flexibility provided by bifocal lenses for manipulating peripheral refractive errors.

One disadvantage of the bifocal design is that the slope of the surface is discontinuous at the boundary between inner and outer zones. This discontinuity is a disadvantage for fabrication of the lens and for achieving robust optical performance across different pupil sizes and different field angles. To avoid these problems, a transition zone is usually incorporated between the inner zone and outer zone for this purpose. We implemented a smooth transition zone by least-square fitting of the lens surface with a set of polynomials up to the 30th order. The RMS of the residual fitting error was 0.08 μm, which is negligible compared to typical fabrication tolerances. Residual refractive errors for this smoothed design (BF Smooth) were indistinguishable from the original bifocal design (BF). Smoothing increased RMS wavefront error slightly along the foveal line-of-sight from 0 (design BF) to 0.07 μm (0.13 wave, design BF smooth) over a 5 mm entrance pupil. Nevertheless, the peripheral optical quality of this design is better than the classic design CV within 45 degree field of view (Figure 2b).

Figure 2.

Performance of customized contact lens that introduces a relative myopic pattern onto myopic eye's peripheral visual field. (a) Spherical refraction pattern of the Navarro myopic eye (solid square), design CV (solid circle), design BF (empty upper-triangle), and design BF Smooth (empty lower-triangle). (b) Root-mean-square (RMS) of wavefront errors (lower- and higher order aberrations) of the Navarro myopic eye (solid square), design CV (solid circle), and design BF Smooth (empty lower-triangle).

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For bifocal lenses, the assumed size of the entrance pupil (EP) along the foveal line-of-sight is an important design parameter for determining the relative sizes of inner and outer zones of the lens. Moreover, the relative dimensions of the two zones affect the balance achieved between the central vision and peripheral vision corrections. The bifocal design BF reported in Table 2 was designed for the 5 mm EP of the myopic model eye. Applying the same OWT procedure to other pupil sizes, the bifocal lens optimized for a 4 mm EP or a 6 mm EP differ significantly from the 5 mm design as reported in Table 3. After smoothing these bifocal designs by polynomial fitting, we calculated their peripheral refractive errors. In each design, the peripheral refractive error varies only slightly with pupil size. Therefore the peripheral refractive error of each design reported in Figure 3a, which was computed at the design pupil size, is representative of all pupil sizes. Figure 3a reveals that the peripheral refractive error patterns of the “4 mm design” is more effective than the “5 mm design” or the “6 mm design” at introducing peripheral relative myopia. This is because the ‘4 mm design’ has a larger outer zone, which manipulates peripheral refractive error more effectively. The penalty of a small inner zone is reduced retinal image quality along the foveal line-of-sight when the actual EP becomes larger than the design size. This penalty is quantified by the modulation transfer functions (MTF) in Figure 3b. The MTF for the 4 mm design is significantly depressed for a 5 mm EP and even more depressed for a 6 mm EP. By comparison, the MTF provided by the 5 mm design is superior when the EP is 5 mm but once again becomes depressed if the EP exceeds the design size (e.g. 6 mm EP, 5 mm design pupil). Therefore, in practice, the relative dimensions of inner and outer zones of the bifocal should be selected carefully to achieve the desired balance between foveal image quality and peripheral refractive errors for habitual pupil sizes.

Figure 3.

Performance of customized bifocal contact lenses designed to introduce relative myopic pattern onto the peripheral visual field (a) Relative peripheral refractive error of the 4 mm, 5 mm, and 6 mm designs when the entrance pupil size matches the design pupil size. (b) Modulation transfer functions (MTF) of foveal corrections for different combinations of entrance pupil sizes and design pupil sizes.

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