OPTICAL CORRECTION

By Professor Mo Jalie

This article explains how aspherical surfaces can be used to improve the off-axis performance of strong plus lenses, such as those used for the correction of aphakia.

Figure 1. Aspherical surface - prolate ellipsoid.

In lens design, the term 'aspherical surface' usually refers to a surface that is rotationally symmetrical but not spherical, such as the ellipsoid illustrated in Figure 1, which would be generated by an ellipse rotating about its major diameter. The ellipsoid is one member of a family of aspherical surfaces known collectively as the conicoids, since they result from the rotation of a conic section about its x-axis. The spherical surface is also a member of the family. An ellipse rotated about its x-axis produces an ellipsoid. If the major axis of the ellipse is horizontal, the solid is referred to as a prolate ellipsoid. If the minor axis is horizontal the solid is referred to as an oblate ellipsoid. When a parabola is rotated about the x-axis it generates a paraboloid and a hyperbola generates a hyperboloid.

CORRECTING APHAKIA WITH ASPHERICS

Figure 2. Field diagram for +12.O0 D lens made with spherical surfaces. Note that for a 20' rotation of the eye the effective Rx is +12.00/+0.50, and at 30* the effective Rx is +11.931+1.37.

The field diagram illustrated in Figure 2 shows the off- axis performance of a +12.00 D lens employing spherical surfaces. The increase in tangential power of the lens and the large amount of aberrational astigmatism can be seen in the diagram. The sagittal power remains about +12.OO D, but the tangential power increases, reaching about +14.00 D at 35" from the optical axis. At 35 ° , the real effect of this lens form with spherical surfaces is +11.91 with a +2.00 cylinder, not the +12.00 sphere intended. At this point, the lens exhibits 2.00 D of unwanted astigmatism. When the designer is not limited to the use of spherical surfaces, oblique astigmatism can be eliminated to provide a big increase in the field of useful vision.

Figure 3. How an ellipsoidal surface corrects aberrational astigmatism. A is the vertex of the curve. C., is the centre of curvature of the surface at the vertex. AC,. is the radius of curvature of the surface at the vertex, r... P is a point on the curve. PC.P is the radius of curvature of the surface at point P in the tangential meridian, which is the plane of the diagram. C5P lies on the evolute, C.,£.B which is the focus of the tangential centres of curvature of the surface between points A and B. PC..P is the radius of curvature of the surface at point P in the sagittal meridian, which lies at right angles to the plane of the diagram. £,P lies on the evolute, C6 C5 B, which is the locus of the sagittal centres of curvature of the surface between points A and B.

This is achieved by employing a surface which itself is astigmatic, the surface astigmatism varying in just the right way to counteract the astigmatism of oblique incidence. One of the simplest surfaces to provide the correct variation in neutralizing astigmatism is the ellipsoid. It is easy to see how such a surface introduces neutralizing astigmatism by considering how the surface alters in shape as the eye rotates away from the pole of  the curve. Figure 3 illustrates the instantaneous centres of curvature for the point P on the surface of a convex prolate ellipsoidal surface. The evolutes for the section AB are also shown and it can be seen that both the tangential and the sagittal radii of curvature for the surface increase, ie the tangential and the sagittal surface powers decrease, with the tangential radius changing faster than the sagittal radius. Inspection of  the field diagram in Figure 2 confirms that a greater decrease in the tangential power of the lens is just what is required to combat the aberrational astigmatism for this form of lens.

When the designer is not limited to the use of spherical surfaces, oblique astigmatism can be eliminated to provide a big increase in the field of useful vision.

By careful choice of eccentricity for the ellipsoid it is possible to eliminate oblique astigmatism for wide zones of the lens. Aspheric lenses of the type needed for the correction of aphakia usually employ a convex prolate ellipsoidal surface to eliminate aberrational astigmatism in the post-cataract range of prescriptions.

Figure 4. Field diagram for +12.00 D lens made with convex prolate ellipsoidal surface. Note that for a 20" rotation of the eye the effective Rx is +11.68 DS, and at 30" the effective Rx is +11.33 DS.

The improvement in off-axis performance can be judged from the field diagram shown in Figure 4, which illustrates the zonal variation in oblique vertex sphere powers for a point-focal +12.00 D lens made with a -3.00 D back curve and a suitably chosen ellipsoidal front surface whose p-value is +0.65. It can be seen for this design that the tangential and sagittal oblique vertex sphere powers remain the same for all zones out to 40", but the lens performance is by no means perfect.

The mean oblique power, which now is the same as the tangential and sagittal oblique vertex sphere powers, drops off rapidly as the eye rotates away from the optical axis of the lens. This loss in power, the mean oblique error, amounts to almost 1.00 D at 35 ° from the optical axis, but at least the error in off-axis performance is a spherical one. It goes without saying that, ideally, the designer would like to be able to increase the marginal power of the aspheric design in order to provide a constant correction for all zones of the lens. The large drop in tangential power does provide the advantage, for lens powers in this range, of a reduction in distortion compared with the spherical design.  

20/20 05/02