Refraction by a Single Curved Surface
The next simplest refracting system is a single curved surface separating regions of different refractive indices. Several simplifying assumptions are appropriate: we assume that the surface is circularly symmetric about a central axis (in this context referred to as the “optic axis”), and that an incident ray of light is coplanar with the optic axis (that is, we ignore “skew” rays). If we assume that the rays of light travel in close proximity to the optic axis and make only small angles with the optic axis (the “paraxial regime”), we may deduce the following 2 equations from Snell’s law, simple geometry, and the “small-angle” trigonometric approximations (Appendix 1.1):
Here n1 and n2 are the refractive indices of the regions to the left and to the right of the refracting interface, u is the (signed) distance from the source object point to the refracting interface, ν is the (signed) distance from the refracting interface to the image of the source object point, and r is the (signed) radius of curvature of the refracting interface. The sign convention for u, ν, and r is that distances to the left of the interface are taken as negative numbers; distances to the right of the interface are taken as positive numbers. Light is assumed to travel from left to right. The quantity P is referred to as the power of the refracting interface.
If either medium is air, the term n1/u or n2/ν reduces to a simple reciprocal—1/u or 1/ν, respectively—and is referred to as the object vergence or image vergence. If the refractive index of either medium is greater than 1.0, the quantity n1/u or n2/ν is referred to as the reduced vergence (even though these quantities are larger than the simple reciprocals). In either case, the vergences (or reduced vergences) are frequently abbreviated as n1/u = U or n2/ν = V, and the vergence equation takes the simple form:
U + P = V
Both the vergences U and V and the refractive power P have the units of reciprocal meters, which in this context is called the diopter, abbreviated D. In many cases, especially regarding the human eye, measurements must always be converted to units of meters for purposes of the vergence equation, although typical distances are more conveniently measured in centimeters or millimeters. The distance equivalents of some common dioptric values are listed in Table 1-2.
Table 1-2 Some Common Dioptric Values and Their Distance Equivalents
This simple geometry, with a single curved refracting surface separating 2 regions of different refractive index, seldom arises in practice. An important exception involves the aphakic eye. Because such an eye has no lens, as a first approximation, we may ignore the interface between the cornea and the aqueous humor, which decreases the refractive power of the air–cornea interface by about 10%. (See Example I-6 in the Quick-Start Guide.)
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.