In addition to the axial refractive errors discussed previously, it is necessary to deal with refracting surfaces that lack circular symmetry, which allows spherical lenses to focus all the light emerging from a point source in a single image point. Consider a “toroidal” surface, such as the side of a rugby ball, an (American) football, or the outer rim of automobile tire inner tube (a torus), as in Figure I-13.
To envision the effect of such a refracting surface, called a toric lens, consider the curvature of the intersection of the surface with a plane that rotates about the line perpendicular to the surface at the apex (the “normal” line; see Figure I-14). The orientation of such a plane, or the intersection curve itself, is referred to as a meridian of the lens.
Figure I-13 The outer rim of a torus forms a toric surface.
(Illustration by Ir. H. Hahn, from Creative Commons.)
Figure I-14 A normal plane intersecting a toric surface. The curvature of the curve where the plane intersects the surface will vary with the orientation of the plane about the perpendicular to the apex. Each such curve is a meridian of the surface at that point.
As this normal plane rotates, the curvature of the intersection varies, from a flattest meridian to a steepest meridian, 90° away (Figure I-15).
Light rays from a distant point source that land on the steepest meridian are refracted as if they encountered a spherical lens of the same curvature. Light rays that land on the flattest meridian are refracted as if they encountered a spherical lens with less dioptric power. Thus, the focal length varies with the choice of meridian, and the toroidal surface does not image the light from a single point source at a single image point. This situation is referred to as astigmatism, from the Latin for “absence” of a (single) point, or focus (Figure I-16).
The flattest and steepest meridians lie 90° apart from each other; they are known as principal meridians. We can keep track of this situation by drawing a power cross, which shows the orientation of the steepest and flattest meridians, and indicates their respective refractive powers (Figure I-17).
In practice, it is often convenient to emphasize the difference in the refractive powers of the principal meridians. In this case, we can interpret the refracting surface as the combination of a spherical refracting surface, which refracts equally in all directions, and a purely “cylindrical” refracting surface, with maximal power corresponding to one of the directions indicated in the power cross (and no refracting power in the orthogonal direction). Thus, for example, a lens with a power of +1.00 D horizontally and + 2.00 D vertically could be described as the combination of a spherical lens with power +1.00 D and a cylindrical lens with power +1.00 D in the vertical direction. This is notated +1.00
+1.00 @ 90°. Alternatively, the same refracting surface could be described as a spherical lens with power +2.00 D combined with a cylindrical lens with power –1.00 D acting in the 180° meridian, notated +2.00
–1.00 @ 180°.
Figure I-15 Variation of the curvature of a toric surface along 2 different meridians. The meridians for which the curvature is minimal or maximal are 90° apart.
Figure I-16 Astigmatic image formation by a toric surface. In this example, light rays that land on the horizontal meridian (ie, rays shown in blue) reach a focus farther from the lens than rays that land on the vertical meridian (ie, rays shown in red).
Figure I-17 Power-cross representation of a toric lens with slightly oblique axes. The meridians with maximal and minimal power are shown.
Indeed, we can realize such lenses in practice by combining spherical lenses with actual lenses with cylindrical surfaces, either convex (“plus cylinders”) or concave (“minus cylinders); see Figure I-18. These can be found in standard trial lens sets.
These cylindrical lenses are marked to indicate the orientation of the original axis of the glass cylinder from which the cylindrical surface was derived. The refracting power of such a cylindrical lens acts in the direction perpendicular to the orientation of the cylinder axis. For example, we obtain a plus cylinder lens with power +1.00 D along the horizontal meridian by slicing the surface from a glass cylinder with axis vertical. Such a glass cylinder lens is denoted +1.00 × 90°. The notation for combinations of spherical and cylindrical lenses uses the “×” symbol (“x” for “axis”). Thus, the lens described in the preceding paragraph can be realized with either a plus cylinder or minus cylinder combination: +2.00
–1.00 × 90° or +1.00
+1.00 × 180°.
Figure I-18 Convex (plus cylinder) and concave (minus cylinder) glass lenses.
Notice that we use the notation “@” (read “at”) to denote the meridian (direction) in which a cylindrical lens exerts refractive power. That contrasts with the “×” notation (read “axis”), which indicates the orientation axis of the cylindrical lens, which will exert its refractive power 90° away from the orientation of the axis.
There are always 2 choices to describe the same toric refractive surface: (1) start with the maximal sphere power and describe the minus cylinder that provides the difference between the powers of the strongest and weakest meridians, or (2) start with the minimal sphere power and describe the plus cylinder that provides the difference between the weakest and strongest meridians. To convert from one description to the other, simply add the sphere and cylinder (keeping track of + and – signs) to determine the new sphere power, change the sign of the cylinder, and add or subtract 90° to the axis (to keep the final axis between 0° and 180°). The process of converting between the plus cylinder and minus cylinder descriptions of a toric lens is referred to as transposition. This duality occurs in every context where we consider toric lenses.
Excerpted from BCSC 2020-2021 series : Section 3 - Clinical Optics. For more information and to purchase the entire series, please visit https://www.aao.org/bcsc.