–1 + (–4) = V V = –5 D
In this case, the image rays are diverging, and a virtual image will appear to be located 20 cm to the right of the mirror. The image is minified and erect.
Consider another example: What is the reflecting power of the anterior surface of the cornea if this surface has a radius of 8 mm? The focal length is 8/2 = 4 mm, the inverse of which is –250 D power. Thus, an object at infinity would appear as an image slightly behind the iris plane, 4 mm posterior to the anterior corneal surface. This example shows why a camera focused on light reflections in the cornea is approximately in focus in the plane of the iris.
Spherocylindrical Lenses
The lenses and mirrors discussed thus far have been radially symmetric about an optical axis, so a flat diagram has sufficed. Now, however, we need to think in 3 dimensions in order to discuss the lenses used to correct regular astigmatism of the eye.
A cylinder has no curvature in one direction and has spherical curvature in the meridian perpendicular to that direction. A spherocylindrical lens has the shape of a torus. That is, its shape is similar to that of the outer surface of a bicycle tire or barrel, with a greater and a lesser circular curvature meeting at right angles where the tire (or the barrel lying on its side) touches the ground—at the point where we would find the optical center of a spherocylindrical spectacle lens (Fig 1-55).
Figure 1-55 Toric surfaces. (Redrawn from Duane TD, ed. Clinical Ophthalmology. Vol 1. Hagerstown, MD: Harper & Row; 1985:47, Fig 3253, with permission from Lippincott Williams & Wilkins.)
We could describe such a spherocylindrical lens as having a power of –3 D at 30° and –5 D at
120°. Alternatively, we could describe the same lens by saying it is a –3 D sphere combined with a – 2 D cylinder lens with the axis of the cylinder placed at 120° (–3.00 –2.00 × 120), or as a –5 D spherical lens combined with a +2 D cylinder lens with its axis held at 30° (–5.00 +2.00 × 30). The “spherical equivalent” lens, halfway between the 2 powers in diopters, is –4 D. To find that, we add half the cylinder amount to the sphere (Fig 1-56).
Figure 1-56 Example of a power cross. (Illustration developed b y Leon Strauss, MD, PhD.)
To convert a prescription from positive cylinder form to negative cylinder form (or conversely), consider the following points:
The new sphere is the algebraic sum of the old sphere and cylinder.
The new cylinder has the same value as the old cylinder but with opposite sign.
The axis needs to be changed by 90°.
Combination of Spherocylindrical Lenses
If we hold 1 of these lenses just in front of a second lens, with the axes at any angle with respect to each other, the result will be another spherocylindrical lens; again the greatest and least curvature will be 90° apart. Determining the new axis and power is simple if the 2 old axes are aligned but is generally too much trouble to calculate by hand otherwise.
The Conoid of Sturm
A pencil of light rays is not brought to a point focus by passing perpendicularly through the center of a spherocylindrical lens; rather, 2 focal lines are formed, 1 for each of the 2 powers of the power cross. The geometric envelope of a pencil of light rays refracted by a circular-aperture spherocylindrical lens is called the conoid of Sturm. This envelope of light rays traveling along the conoid of Sturm of a lens, for which both powers are positive, has an elliptical cross section that collapses first to 1 focal line and later to the other. The average of the 2 diopter lens powers is the spherical equivalent, and only at its place along the conoid of Sturm is the cross section circular, at which point it is called the circle of least confusion (Fig 1-57).
Figure 1-57 Conoid of Sturm. (Illustration developed b y Kevin M. Miller, MD, and rendered b y Jonathan Clark.)