Figure 1-23 Light traveling the shorter distance from object (O) to image (I) point traverses a thick part of the lens. Light traveling the longer path 2 goes through less glass. If the lens is properly shaped, the greater distance in air is perfectly compensated for by the shorter distance in glass, and the time required to travel from object to image is identical for both
paths. (Illustration developed b y Edmond H. Thall, MD, and Kevin M. Miller, MD, and rendered b y C. H. Wooley.)
Pinhole Imaging
The pinhole camera is the earliest (c. 400 BC) known imaging device. Creating your own pinhole camera is an easy and worthwhile exercise. You can make a viewing screen by taping a piece of waxed paper to a simple frame cut out of poster board or the backing of a pad of paper. Another method is to create a pinhole near the middle of a fairly large opaque material such as a large index card.
In a dark room, light a candle, hold the pinhole about 30 cm (≈1 foot) from the candle, and place the screen about 30 cm behind the pinhole. You should observe an inverted image of the flame on the screen.
There is an image anywhere behind the pinhole, so a pinhole camera requires no focusing. However, the image is often too faint to be observed. Increasing pinhole size brightens but also blurs the image.
If you replace the pinhole with a +8.00 D spherical convex trial lens, the image is much brighter but appears in only one location behind the lens. In fact, you will probably have to adjust the distance between the lens and screen to get a clear image. Lenses overcome the main disadvantage of pinhole imaging—faint images—but sacrifice the main advantage—no need to focus. For lenses, image location is crucial.
Locating the Image: The Lensmaker’s Equation
Increase the distance between the candle and lens to about 1 m (≈1.1 yard). You must move the screen closer to the lens to see a brighter, smaller image. Move the lens closer to the candle and the image is farther away from the lens.
The situation is shown schematically in Figure 1-24. The optical axis is an imaginary but welldefined line determined by rotational symmetry of the lens. The vertices V and V′ are the intersections
of the axis with the lens surfaces. More important, the principal points P and P′ are major reference points used to define several other variables. Point P is the object principal point and P′ the image principal point. Object distance, o, is measured from P to the object, and image distance, i, is measured from P′ to the image (Fig 1-25). Note that the principal points are not the same as the vertices and do not even have to be “inside” the lens.
Figure 1-24 The optical axis is an imaginary but well-defined line defined by the lens’s symmetry. The vertices V and V′ are the points of intersection of the axis with the lens surfaces. In general, the principal points P and P′ do not coincide
with the vertices. (Illustration developed b y Edmond H. Thall, MD.)
Figure 1-25 Definition of the variables in the lensmaker’s equation. Object distance, o, is measured along the axis from P to the object, and image distance, i, is measured along the axis from P′ to the image. The positive direction is left to right.
(Illustration developed b y Edmond H. Thall, MD.)
The image location can be calculated using the lensmaker’s equation (discussed below):
By convention, light travels from left to right, which is the positive direction. Suppose an object is 0.50 m in front of a +6.00 D lens. Because object distance is measured from P to the object, its direction is right to left, or negative. According to the lensmaker’s equation,
The unit diopter is a reciprocal meter, so 
diopters
Again, because a diopter is a reciprocal meter, 
meter
i′ = +0.25 m
The lensmaker’s equation
The lensmaker’s equation (LME) is as follows:
where the ratio n/o is the reduced object vergence (U) and the ratio n′/i is the reduced image vergence (V). The concepts of vergence and reduced vergence are discussed in detail in the section Ophthalmic Lenses.
The LME is one of the most important equations in ophthalmology. Unfortunately, it is also one of the most misused equations in ophthalmology.
Fundamentally, the LME says 2 things. First, the location of the image depends on the location of the object. Consider a specific example wherein the refractive index of a glass rod is 1.5 and the radius of curvature is 0.1 m. Suppose an object is in air with n = 1.0. The LME becomes
or
Note the units of reciprocal, or inverse, meters. Suppose the object is 1 m in front of the lens. Object distances are negative, so
Thus, the image is 37.5 cm behind the refracting surface.
Second, the LME establishes a relationship between the shape of the refracting surface and its optical function. The radius of the spherical refracting surface affects the image characteristics. The refractive power (or simply power) of a spherical refracting surface is
To demonstrate the significance of power, consider 2 spherical refractive surfaces, both constructed from glass rods (n = 1.5). Suppose that 1 refracting surface has a radius of 10 cm, as in the previous example, and the other has a radius of 20 cm. If an object is 1 m in front of each surface, where is the image? As shown in the previous example, the first surface has a power of 5.0 D and produces an image 37.5 cm behind the surface. The second surface has a power of 2.5 D and forms an image 1 m behind the refracting surface. Notice that the second surface has half the power, but the image is more than twice as far behind the refracting surface.
Refractive power, strictly speaking, applies to spherical surfaces, but the cornea is not spherical. In general, every point on an aspheric surface is associated with infinitely many curvatures. There is no such thing as a single radius of curvature. The sphere is a very special case: a single radius of curvature characterizes the entire sphere. A single radius can characterize no other shape, and refractive power should not be applied to a nonspherical surface.
In addition, power is a paraxial concept; thus, it applies only to a small area near the optical axis. Power is not applicable to nonparaxial regions of the cornea. In the paraxial region, imaging is stigmatic (ie, paraxial rays focus to a common point). Even for spherical surfaces, rays outside the paraxial region do not focus to a single point. That is, away from the paraxial region, rays do not focus as predicted when the LME is used.
For further information about first-order optics and the lensmaker’s equation, see Appendix 1.2 at the end of the chapter.