Development of freeform machining

Before 1980, aspherical surfaces were only used on high-power plus spectacle lenses that lie beyond the range of powers that can be corrected for aberrational astigmatism with spherical surfaces. In recent years, aspherical surfaces have been employed on lenses of low power; those used for the usual range of prescriptions. Once again this was made possible by the development of freeform machining.

Aspheric lenses for the normal power range

In 1980, the author obtained patents¹ for lenses in the power range +7.00 to -20.00 for a series of spectacle lenses which incorporate a hyperboloidal curve for the major surface of the lens. The major surface being the convex surface for plus powers and the concave surface for minus lenses. The use of aspheric forms for the low to medium power range allows the production of thinner and lighter lenses for the normal range of prescriptions. The reduction in thickness is the result of a two-stage process.

Stage one

First, the lens is made much flatter in form by employing a shallower base curve. Simply by flattening the lens form a saving in center thickness is obtained. The flatter the lens, the thinner it becomes.

If the lens is made with a -1.50 base curve instead of the usual -5.25 inside curve which would be necessary to make the lens free from astigmatism (i.e., point focal), then a saving in center thickness of 0.6 mm would be obtained. Needless to say, this flatter form does not have good optical properties. It is afflicted with positive aberrational astigmatism when the eye rotates to view through off-axis portions of the lens as shown by the field diagram in Figure 1b.

Just how poor the off-axis performance becomes due to flattening the lens form is illustrated in figure 1. Figure 1a illustrates a field diagram for a +4.00 D lens made with a -5.25 back curve and it can be seen that the tangential and sagittal oblique vertex sphere powers are the same for all direction of gaze.

This form is free from oblique astigmatism and represents a point-focal form for this power. Figure 1b illustrates the off-axis performance of a +4.00 D lens made with a -1.50 back surface power using spherical surfaces and it is seen that the real effect of the lens when the eye has rotated 35° from the optical axis is +4.05/+0.87. It will be appreciated that there is almost 1.00 D of aberrational astigmatism 35° from the optical axis for this very shallow bending.

However, to eliminate the aberrational astigmatism an aspherical surface can be employed whose form is such that it introduces negative surface astigmatism to neutralize the astigmatism of oblique incidence. A correctly chosen aspherical surface will completely neutralize the aberrational astigmatism arising from oblique incidence.

Figure 1c illustrates the off-axis performance of the +4.00 D lens made with a -1.50 back surface power and a convex aspherical surface whose p-value has been chosen to neutralize the astigmatism of oblique incidence. This form has the same oblique vertex sphere powers as the point-focal form with spherical surfaces whose performance is depicted in figure 1a. The surface is a convex hyperboloid whose p-value is -1.8 and it can be seen that the field diagram is almost identical with that shown in figure 1a for the spherical form.

Stage two

The second stage of the thinning process occurs since, for a given diameter, the required aspherical surface has a smaller sag than a spherical surface of the same vertex radius. The smaller front surface sag causes a further reduction in the center thickness of the lens.

The original patent proposed that a hyperboloid should be employed for the major surface of the lens since the rate of flattening of a hyperboloid is just what is required to neutralize aberrational astigmatism.

Figure 2 shows what additional saving in center thickness is achieved when the convex spherical surface is replaced by a suitable convex hyperboloidal surface whose asphericity is chosen to restore the off-axis performance of the lens. A further saving of 0.6mm is achieved for a 70mm diameter when the spherical surface is aspherised to eliminate the aberrational astigmatism arising from oblique incidence.

The aspheric lens form has a total saving in center thickness of 1.2mm when compared with the traditional spherical form. Needless to say, any higher order aspherical surface could be used but, in practice, it would not depart significantly from a hyperboloid since this curve regulates the astigmatism at the correct rate.

The optical performance of an aspheric design can be made to match any design philosophy. The lens may be made point-focal, just like the designs illustrated in figures 1a and 1c or it may be made in Percival form or, more typically, a compromise bending between these two forms to provide a reasonable performance over a wide range of fitting distances.

An even greater saving in thickness is obtained when a higher refractive index material is used. If the same power base curve is used the saving is two-fold. Firstly, there is the obvious reduction in the sags of the curves since longer radii of curvature are employed.

Secondly, since the use of the same power base curve on a higher refractive index material requires a longer radius of curvature at the vertex, r0, effectively, the lens is flatter still and requires greater asphericity on the convex surface to restore the off-axis performance. This is illustrated in figure  3, which shows how the center thickness of a 70mm diameter +4.00 D lens would reduce when made in 1.60 and 1.70 index materials. The asphericity of the convex surfaces indicated in the figure has been chosen to provide the same off-axis performance for each lens.

Another important advantage of these low-power aspheric designs for hypermetropia can be gleaned from figure 3. The original best-form +4.00 design with spherical surfaces required a center thickness of 6.6mm in order to obtain an edge thickness of 1.0mm at 70mm diameter. If this uncut lens is edged down to a finished diameter of 50mm, it will have an edge thickness of 4.1mm which is not acceptable for a lens of this power.

The aspheric design made in 1.60 index material, on the other hand, has a center thickness of  4.5mm and would have an edge thickness of 2.6mm when edged down to a finished diameter of 50mm. The aspheric design lends itself far better to a system of supply of large diameter plus uncut lenses which need to be edged to smaller diameters depending upon the choice of shape and size of the lens.

Aspheric lenses for myopia

The principle of flattening a curved lens form to make it thinner and then aspherising one surface to restore the off-axis performance of the flatter-form lens can be applied equally to minus lenses. For example the reduction in thickness which is obtained for -4.00D lenses made in CR 39 material with uncut diameters of 70mm and center thickness of 2.0mm is shown in the figure 4.

It can be seen that the traditional best form design made using spherical surfaces might employ a +4.75 D base curve, when the resulting edge thickness would be 8.0mm. Then, flattening the base curve to +0.75 D produces an edge thickness of 7.1mm which is a saving of 0.9mm at the edge.

Finally, aspherising the flatter-form lens to provide the same off-axis performance as the best-form spherical design results in an edge thickness of 6.4mm which is a further saving of 0.7mm, the final aspheric design being 1.6mm thinner than the traditional spherical form.

The author’s original proposal for the correction of myopia was to employ a concave hyperboloidal surface but, initially, lens manufacturers preferred to aspherise the convex surface of the lens since it is easier to incorporate the cylinder on the concave surface as a minus-base toric. Several aspheric minus lens series, therefore, incorporate a convex aspherical surface, the purpose of which is to increase the convexity of the front surface towards the edge of the lens (Fig. 5). Typically, a convex oblate ellipsoid might be used whose tangential curvature increases at a faster rate than that of a spherical surface of the same vertex radius, as illustrated in figure 5a.

Usually, however, a two or three-term polynomial convex curve is chosen since this does not place a restriction on the maximum diameter of the lens. A field diagram for a typical minus aspheric lens with a convex polynomial surface is illustrated in figure 5b. 

Freeform machining now enables the concave surface of the lens to be aspherised enabling the lens to incorporate the astigmatic correction together with the aspherical surface.  

For higher power minus lenses, the principle of blending has been applied to the humble workshop flattened lenticular to produce a blended concave lenticular with a truly invisible dividing line. These blended lenticulars for myopia, such as the Wrobel Super-lentiand the Rodenstock Lentilux designs, enjoy excellent cosmetic properties and allow very high minus prescriptions, in excess of -20.00 D, to be dispensed in relatively thin and lightweight form.

Atoric lenses

Aspherical surfaces of the type which have been described so far provide excellent imaging properties for any lens power, providing that the prescription is spherical. It should be apparent that in the case of astigmatic prescriptions, the asphericity of, say, a conicoidal surface can only be correct for one principal meridian of the lens. The other principal meridian will require a different eccentricity, or different p-value for the power along this meridian².

For example, in the case of the prescription +2.00/+2.00 x 180, which has been made as an aspheric lens with a -1.50 base curve, the principal meridians of the lens have powers of +4.00 and +2.00. It has already been pointed out that the +4.00 meridian which requires a -1.50 base curve would be point-focal if the front curve had a hyperbolic section with a p-value of -1.8. Accurate trigonometric ray-tracing shows that the +2.00 meridian with a cross-curve of -3.50 would need a p-value of +0.45 if this meridian is also to remain point-focal for the 35° zone of the lens.

Such a surface is depicted in figure 6 which illustrates a convex atoroidal surface whose “toricity” is due to a change in asphericity from one meridian to a second meridian at 90° to the first. It should be understood that the surface illustrated in figure 6 has no cylindrical power in the usual sense of the term since the curvatures of the surface at the vertex along the two principal meridians are identical. The cylindrical component of the lens is provided in the usual way by grinding a toroidal surface on the back of the lens. One might argue that the term atoroidal is not really a good description for this type of surface and a better definition for the surface might be a non-rotationally symmetric aspherical surface. However, this term also describes many other forms of surface, including progressive power surfaces, and the term atoroidal seems to have entered the literature.

Such a surface was employed for the original Zeiss Hypal series of lenses when employed for astigmatic prescriptions. For spherical prescriptions the convex surface of the Hypal design is simply an aspherical surface which is virtually indistinguishable from a conicoid. A true atoroidal surface whose principal vertex curvatures differ by the required cylindrical component in addition to a variation in asphericity for the two principal meridians is also employed on modern spectacle lens forms such as the Pentax Super Atoric 1.67 UV AR and the Seiko SSV AZ 1.67 Bi-aspheric design (Fig. 7).

References: 1. US Patent 4289387 (1981) Jalie M., Ophthalmic Spectacle Lenses having a Hyperbolic Surface, 2. US Patent 5083859 (1992), Jalie M., Aspheric lenses.