posterior principal point to the posterior focal point.

Following the sign convention, focal lengths are negative when the focal point is to the left of the principal point and positive when the focal point is to the right of the principal point. For instance, a +5 D thin lens in air has an AFL of –20 cm and a PFL of +20 cm.

For any optical system, focal lengths and refractive power P are related by

For any optical system, the distance from the anterior principal point to the anterior nodal point is always equal to the distance from the posterior principal point to the posterior nodal point. The distance between principal point and nodal point follows the sign convention and is given by

Distance = AFL + PFL

For instance, for a +5 D thin lens in air, AFL + PFL = –20 cm + 20 cm = 0. Thus, the nodal points and principal points overlap. For a +5 D thin lens with water (n = 1.33) in front and air in back, the AFL = –26.6 cm and the PFL = 20 cm. Thus, the nodal points are 6.6 cm to the left of the principal points.

Gaussian Reduction

Thus far, we have discussed the properties of a single optical system. The treatment of refractive errors usually involves adding a lens to an existing optical system, the patient’s eye. Gaussian reduction describes what happens when 2 optical systems (such as a correcting lens and the eye) are combined.

When 2 optical systems—each with its own cardinal points—are combined, a totally new optical system is created that is described by a new set of cardinal points. The thick-lens equation is used to reduce the 2 individual systems to a single system with its own set of cardinal points. Typically, the combined system’s cardinal points and power differ from those of either of the individual systems. Clinically, Gaussian reduction is most important when used in conjunction with the correction of ametropias (discussed in Chapter 3) and in the calculation of IOL power (see Chapter 5).

Knapp’s Law, the Badal Principle, and the Lensmeter

One problem in treating refractive errors is that the correcting lens often changes the size of the retinal image. If the retinal image in 1 eye differs in size from that in the other eye, the difference is usually tolerated by the patient unless this difference is large. The adult brain can fuse retinal images that differ in size by as much as 8%; the child’s brain can handle an even greater disparity. According to Knapp’s law, the size of the retinal image does not change when the center of the correcting lens (to be precise, the posterior nodal point of the correcting lens) coincides with the anterior focal point of the eye (Fig 1-41).