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

Abstract

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.

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

“…The rotationally asymmetric patterns of total astigmatism magnitude (bow-tie centered on the estimated optical axis) in Figure 1 is expected theoretically when oblique astigmatism interacts with axial astigmatism (Liu & Thibos, 2016). In this case, theory predicts total astigmatism maps with two null points along the meridian perpendicular to the axial astigmatism axis.…”

“…Theoretically, in situations where a paraxial approximation of the system is sufficient, the optical axis may be defined as the path of collinear entrance and exit rays (Harris, 2009). For present purposes, however, the most useful definition is a line passing through the eye's entrance pupil that is coincident with the axis of rotational symmetry for the oblique component of ocular astigmatism (Liu & Thibos, 2016). …”

“…We take advantage of that fact to define the system's axial astigmatism as that value of ocular astigmatism found (usually by interpolation) on the optical axis. We then computed a visual field map of oblique astigmatism J oblique by performing a power vector subtraction (Liu & Thibos, 2016) of axial astigmatism J axial from total astigmatism J total for every field location, using the same vector equation, ocular, corneal and internal astigmatisms were computed from Zernike aberration coefficients and then decomposed into axial and oblique component for evaluation. …”

“…Then the numerical ray tracing was performed along the same scanning pattern carried out in our ocular aberration experiment. The resultant Zernike coefficients for corneal aberrations were scaled appropriately to match the eye's pupil size during ocular aberration measurements (which varied with accommodation), and the results expressed as power vectors for corneal astigmatism as described previously (Liu & Thibos, 2016). …”

“…Unlike oblique astigmatism, the magnitude and axis of axial astigmatism are independent of field angle. When astigmatism is represented by a power vector in Cartesian form (Thibos, Wheeler, & Horner, 1997), optical theory predicts that total astigmatism is the vector sum of these oblique and axial forms of astigmatism (Liu & Thibos, 2016). This suggests that decomposing total astigmatism into its axial and oblique components will lead to a clearer understanding of the nature of ocular astigmatism of the human eye and the influence of accommodation.…”