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R. Navarro; Objective Refraction From the Gradient and Curvature of the Wavefront. Invest. Ophthalmol. Vis. Sci. 2010;51(13):3942.
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© ARVO (1962-2015); The Authors (2016-present)
The problem of measuring the objective refractive error of an eye with an aberrometer has shown to be elusive. The purpose of this work is to develop a theoretical framework to determine the refractive error directly from the first and second derivatives of the wavefront.
In a previous work [Navarro, J. Biomed. Opt. 14, 024021, 2009] the vergence error of a ray, passing through a given point (x, y) at the pupil, and intercepting the retina at another point (X’,Y’), was formulated as a 2x2 symmetric matrix: The matrix elements are the ratios between image and pupil coordinates (X’/x, Y’/y, etc.) X’ and Y’ are partial derivatives of the wavefront. Here an equivalent but more robust formulation is presented in terms of wavefront curvature. Curvature can be approximated by the second fundamental quadratic form of differential geometry. This form is another 2x2 symmetric matrix whose elements are proportional to the second derivatives of the wavefront. For both magnitudes vergence and curvature, the 2x2 matrix represents an infinitesimal elliptical wavefront (around each ray) characterized by a refractive error (sphere, cylinder and axis.)
Vergence error and curvature provide quite similar measures of refractive error of a ray (or infinitesimal patch of a wavefront), but vergence error has singularities at the x and y axes, whereas curvature is well defined at any point inside the pupil. These magnitudes show interesting invariant properties against odd-symmetric aberrations, both for the chief ray and for the pupil average. In particular, wavefront curvature of the chief ray provides the refractive error of the paraxial image. A refractive correction producing a nearly maximum contrast (Strehl ratio) is obtained by cancelling either vergence or curvature for the highest number of rays.
The proposed formulation of refractive error as a 2x2 matrix permits a direct link with both differential geometry of the wavefront (curvature) and geometrical optics (vergence error). One can choose different possible corrections and make approximated estimations of image quality and contrast.
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