{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T20:53:31Z","timestamp":1767905611940,"version":"3.49.0"},"reference-count":21,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,5,2]],"date-time":"2023-05-02T00:00:00Z","timestamp":1682985600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,5,2]],"date-time":"2023-05-02T00:00:00Z","timestamp":1682985600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Sci Rep"],"abstract":"Abstract<\/jats:title>To provide a solution for average paraxial lens power (Ap<\/jats:sub>P) of a lens. Orthogonal and oblique sections through a lens of power $$F$$<\/jats:tex-math>\n F<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> were reduced to a paraxial representation of lens power followed by integration. Visual acuity was measured using lenses of different powers (cylinders of \u2212\u00a01.0 and \u2212\u00a02.0D) and axes, mean spherical equivalent (MSE) of S\u2009+\u2009C\/2, Ap<\/jats:sub>P and a toric correction, with the order of correction randomised. A digital screen at 6\u00a0m was used on which a Landolt C with crowding bars was displayed for 0.3\u00a0s before vanishing. The general equation for a symmetrical lens of refractive index (n), radius of curvature R, in medium of refractive index n1, through orthogonal ($$\\theta$$<\/jats:tex-math>\n \u03b8<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>) and oblique meridians ($$\\gamma$$<\/jats:tex-math>\n \u03b3<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>) as a function of the angle of incidence ($$\\alpha$$<\/jats:tex-math>\n \u03b1<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>) reduces for paraxial rays ($$\\alpha \\sim 0$$<\/jats:tex-math>\n \n \u03b1<\/mml:mi>\n \u223c<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>) to $$F_{n,R} \\left( {\\alpha ,\\theta ,\\gamma } \\right)\\left. \\right|_{\\alpha \\sim 0} = \\frac{{n - n_{1} }}{R}\\cos^{2} \\theta \\cos^{2} \\gamma$$<\/jats:tex-math>\n \n \n F<\/mml:mi>\n \n n<\/mml:mi>\n ,<\/mml:mo>\n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n \n \u03b1<\/mml:mi>\n ,<\/mml:mo>\n \u03b8<\/mml:mi>\n ,<\/mml:mo>\n \u03b3<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mfenced>\n \n \n \n \u03b1<\/mml:mi>\n \u223c<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n =<\/mml:mo>\n \n \n n<\/mml:mi>\n -<\/mml:mo>\n \n n<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n R<\/mml:mi>\n <\/mml:mfrac>\n \n cos<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:msup>\n \u03b8<\/mml:mi>\n \n cos<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:msup>\n \u03b3<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The average of this function is $$F_{n,R} \\left( {\\alpha ,\\theta ,\\gamma } \\right)\\left. \\right|_{\\alpha \\sim 0} = \\frac{{n - n_{1} }}{4R} $$<\/jats:tex-math>\n \n \n F<\/mml:mi>\n \n n<\/mml:mi>\n ,<\/mml:mo>\n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n \n \u03b1<\/mml:mi>\n ,<\/mml:mo>\n \u03b8<\/mml:mi>\n ,<\/mml:mo>\n \u03b3<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mfenced>\n \n \n \n \u03b1<\/mml:mi>\n \u223c<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n =<\/mml:mo>\n \n \n n<\/mml:mi>\n -<\/mml:mo>\n \n n<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n \n 4<\/mml:mn>\n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mfrac>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> providing a solution of $$\\frac{F}{4}$$<\/jats:tex-math>\n \n F<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for Ap<\/jats:sub>P.For central (p<\/jats:italic>\u2009=\u20090.04), but not peripheral (p<\/jats:italic>\u2009=\u20090.17) viewing, correction with Ap<\/jats:sub>P was associated with better visual acuity than a MSE across all tested refractive errors (p<\/jats:italic>\u2009=\u20090.04). These findings suggest that $$\\frac{F}{4}$$<\/jats:tex-math>\n \n F<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> may be a more inclusive representation of the average paraxial power of a cylindrical lens than the MSE.<\/jats:p>","DOI":"10.1038\/s41598-023-34010-4","type":"journal-article","created":{"date-parts":[[2023,5,2]],"date-time":"2023-05-02T11:02:20Z","timestamp":1683025340000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Average paraxial power of a lens and visual acuity"],"prefix":"10.1038","volume":"13","author":[{"given":"Stephen B.","family":"Kaye","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jamila","family":"Surti","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"James S.","family":"Wolffsohn","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,5,2]]},"reference":[{"issue":"4","key":"34010_CR1","doi-asserted-by":"publisher","first-page":"382","DOI":"10.1097\/OPX.0b013e31819895b8","volume":"86","author":"SB Kaye","year":"2009","unstructured":"Kaye, S. 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