Modal estimation of wavefront phase from slopes over elliptical pupils

Wide-field optical eye models for emmetropic and myopic eyes

Ocular wavefront aberrations are used to describe retinal image formation in the study and modeling of foveal and peripheral visual functions and visual development. However, classical eye models generate aberration structures that generally do not resemble those of actual eyes, and simplifications such as rotationally symmetric and coaxial surfaces limit the usefulness of many modern eye models. Drawing on wide-field ocular wavefront aberrations measured previously by five laboratories, 28 emmetropic (-0.50 to +0.50 D) and 20 myopic (-1.50 to -4.50 D) individual optical eye models were reverse-engineered by optical design ray-tracing software. This involved an error function that manipulated 27 anatomical parameters, such as curvatures, asphericities, thicknesses, tilts, and translations-constrained within anatomical limits-to drive the output aberrations of each model to agree with the input (measured) aberrations. From those resultant anatomical parameters, three representative eye models were also defined: an ideal emmetropic eye with minimal aberrations (0.00 D), as well as a typical emmetropic eye (-0.02 D) and myopic eye (-2.75 D). The cohorts and individual models are presented and evaluated in terms of output aberrations and established population expectations, such as Seidel aberration theory and ocular chromatic aberrations. Presented applications of the models include the effect of dual focus contact lenses on peripheral optical quality, the comparison of ophthalmic correction modalities, and the projection of object space across the retina during accommodation.

Retinal defocus in myopes wearing dual-focus zonal contact lenses

Purpose

To evaluate the refractive impact of dual‐focus (DF) myopia control contact lenses (CLs) on accommodating young myopic adults.

Methods

Phase 1: accommodative accuracy was assessed in 40 myopic participants. Phase 2: a subset of four subjects who demonstrated accurate accommodation and six who chronically underaccommodated were fitted with single vision (SV, Proclear 1 day) and centre‐distance DF myopia control CLs (MiSight 1 day) with approximately +2.00 D of additional power in two surrounding annular zones. While binocularly viewing high contrast characters at 4.00, 1.00, 0.50, 0.33, 0.25 and 0.20 m, aberrometry data were captured across the central ±30° of the horizontal retina. Local refractive errors were pooled for each area of the pupil covered by the central distance or first annular defocus zone of the DF CLs.

Results

In the “good” accommodator group fitted with SV CLs, accommodative lags were generally absent except at the closest viewing distance (mean errors: −0.09 ± 0.22 D, −0.12 ± 0.26 D, −0.05 ± 0.37 D and +0.38 ± 0.54 D for −2.00, −3.00, −4.00 and −5.00 D target vergences, respectively) but significantly larger in the “poor” accommodating participants (+0.81 ± 0.21 D, +0.97 ± 0.27 D, +1.18 ± 0.39 D, +1.47 ± 0.55 D). For most viewing distances, hyperopic defocus observed in the region of the pupil covered by the first annular zone was replaced with myopic defocus when fitted with the DF CLs. Myopic defocus created by the first annular region was present across the central 30° of the retina.

Conclusions

Some young adult myopes chronically experience high levels of hyperopic defocus when viewing near targets, which was replaced by myopic defocus in the annular part of the pupil covered by the treatment zones when fitted with a centre‐distance myopia control DF CL.

Average gradient of Zernike polynomials over polygons

Wavefront estimation from slope sensor data is often achieved by fitting measured slopes with Zernike polynomial derivatives averaged over the sampling subapertures. Here we discuss how the calculation of these average derivatives can be reduced to one-dimensional integrals of the Zernike polynomials, rather than their derivatives, along the perimeter of each subaperture. We then use this result to derive closed-form expressions for the average Zernike polynomial derivatives over polygonal areas, only requiring evaluation of polynomials at the polygon vertices. Finally, these expressions are applied to simulated Shack-Hartmann wavefront sensors with 7 and 23 fully illuminated lenslets across a circular pupil, with their accuracy and calculation time compared against commonly used integration methods.

Customized models of ocular aberrations across the visual field during accommodation

We aimed to create individual eye models that accurately reproduce the empirical measurements of wave-front aberrations across the visual field at different accommodative states, thus providing a mechanistic explanation for the changes in the eye's aberration structure due to accommodation. Structural parameters of a generic eye model were optimized using optical design software to account for published measurements of wave-front aberrations measured for 19 individuals at 37 test locations over the central 30°-diameter visual field at eight levels of accommodative demand. Biometric data for individual eyes were used as starting values and normative data were used to constrain optimizations to anatomically reasonable values. Customizations of the accommodating eye model accurately accounted for ocular aberrations over the central 30° of visual field with an averaged root mean square fitting error typically below 0.2 μm at any given field location. Optimized structural parameters of the eye models were anatomically reasonable and changed in the expected way when accommodating. Accuracy for representing spherical aberration was significantly improved by relaxing anatomical constraints on the anterior surface of the lens to compensate for not including gradient-index media. Use of the model to compute pan-retinal image quality revealed large penalties of accommodative lag for activating photoreceptor responses to the retinal image.

Variation of axial and oblique astigmatism with accommodation across the visual field

In this study we investigated the impact of accommodation on axial and oblique astigmatism along 12 meridians of the central 30° of visual field and explored the compensation of corneal first-surface astigmatism by the remainder of the eye's optical system. Our experimental evidence revealed no systematic effect of accommodation on either axial or oblique astigmatism for two adult populations (myopic and emmetropic eyes). Although a few subjects exhibited systematic changes in axial astigmatism during accommodation, the dioptric value of these changes was much smaller than the amount of accommodation. For most subjects, axial and oblique astigmatism of the whole eye are both less than for the cornea alone, which indicates a compensatory role for internal optics at all accommodative states in both central and peripheral vision. A new method for determining the eye's optical axis based on visual field maps of oblique astigmatism revealed that, on average, the optical axis is 4.8° temporal and 0.39° superior to the foveal line-of-sight in object space, which agrees with previous results obtained by different methodologies and implies that foveal astigmatism includes a small amount of oblique astigmatism (0.06 D on average). Customized optical models of each eye revealed that oblique astigmatism of the corneal first surface is negligible along the pupillary axis for emmetropic and myopic eyes. Individual variation in the eye's optical axis is due in part to misalignment of the corneal and internal components that is consistent with tilting of the crystalline lens relative to the pupillary axis.

Interaction of axial and oblique astigmatism in theoretical and physical eye models

The interaction between oblique and axial astigmatism was investigated analytically (generalized Coddington's equations) and numerically (ray tracing) for a theoretical eye model with a single refracting surface. A linear vector-summation rule for power vector descriptions of axial and oblique astigmatism was found to account for their interaction over the central 90° diameter of the visual field. This linear summation rule was further validated experimentally using a physical eye model measured with a laboratory scanning aberrometer. We then used the linear summation rule to evaluate the relative contributions of axial and oblique astigmatism to the total astigmatism measured across the central visual field. In the central visual field, axial astigmatism dominates because the oblique astigmatism is negligible near the optical axis. At intermediate eccentricities, axial and oblique astigmatism may have equal magnitude but orthogonal axes, which nullifies total astigmatism at two locations in the visual field. At more peripheral locations, oblique astigmatism dominates axial astigmatism, and the axes of total astigmatism become radially oriented, which is a trait of oblique astigmatism. When eccentricity is specified relative to a foveal line-of-sight that is displaced from the eye's optical axis, asymmetries in the visual field map of total astigmatism can be used to locate the optical axis empirically and to estimate the relative contributions of axial and oblique astigmatism at any retinal location, including the fovea. We anticipate the linear summation rule will benefit many topics in vision science (e.g., peripheral correction, emmetropization, meridional amblyopia) by providing improved understanding of how axial and oblique astigmatism interact to produce net astigmatism.

Specifying peripheral aberrations in visual science

PURPOSE

Investigations of foveal aberrations assume circular pupils. However, the pupil becomes increasingly elliptical with increase in visual field eccentricity. We address this and other issues concerning peripheral aberration specification.

METHODS

One approach uses an elliptical pupil similar to the actual pupil shape, stretched along its minor axis to become a circle so that Zernike circular aberration polynomials may be used. Another approach uses a circular pupil whose diameter matches either the larger or smaller dimension of the elliptical pupil. Pictorial presentation of aberrations, influence of wavelength on aberrations, sign differences between aberrations for fellow eyes, and referencing position to either the visual field or the retina are considered.

RESULTS

Examples show differences between the two approaches. Each has its advantages and disadvantages, but there are ways to compensate for most disadvantages. Two representations of data are pupil aberration maps at each position in the visual field and maps showing the variation in individual aberration coefficients across the field.

CONCLUSIONS

Based on simplicity of use, adequacy of approximation, possible departures of off-axis pupils from ellipticity, and ease of understanding by clinicians, the circular pupil approach is preferable to the stretched elliptical approach for studies involving field angles up to 30 deg.

Designing contact lenses for a wide field of view via ocular wavefront tomography

Purpose

Correcting the off-axis wavefront aberration is potentially important for peripheral vision, for diagnostic imaging of the retina, and for influencing refractive development. A new technique called ocular wavefront tomography (OWT) was adapted to optimize the design of contact lenses to improve the eye's peripheral optical quality.

Methods

OWT is a technique for customizing a multi-surface model eye to mimic the off-axis wavefront aberrations for an individual eye. This technique was adapted for contact lens design by establishing clear design goals for the eye + contact lens system. To demonstrate the method we optimized the shape of an aspheric and bifocal contact lens to correct a wide angle model eye with −2D foveal myopia. Two strategies for correction reflected alternative design goals: 1) to fully correct central vision while also improving optical quality peripherally to enhance vision and retinal imaging, or 2) fully correct central vision while introducing a degree of peripheral myopia relative to central vision in order to slow myopia progression.

Results

The OWT technique successfully produced aspheric and bifocal contact lens designs over a wide field of view. In addition to correcting foveal vision, the optimized contact lens designs either 1) improved the retinal image quality across the visual field (< 45°) significantly to obtain a visual performance and retinal imaging benefit or 2) produced the desired level of myopia in the peripheral field to obtain a refractive development benefit.

Conclusion

The OWT technique is a validated tool to optimize contact lens design over a wide field.