From trigonometry, we can estimate the angle at which light rays from the trabecular meshwork strike the air–tear-film interface. The situation is illustrated in Figure 1-19 using average anatomical dimensions. We ignore the effect of the back surface of the cornea because this surface has relatively little power and we are performing only a rough calculation. From basic trigonometry,

Interestingly, this rough calculation shows that θc is exceeded by only a few degrees. When the cornea is ectatic (as in some cases of keratoconus), the angle of incidence is less than θc and the angle structures are visible without a gonioscopy lens.

Figure 1-19 Average anatomical dimensions of the anterior segment. (Illustration developed b y Kevin M. Miller, MD,

and rendered b y C. H. Wooley.)

Dispersion

With the exception of a vacuum, which always has a refractive index of 1.000, refractive indices are not fixed values. They vary as a function of wavelength. In general, refractive indices are higher for short wavelengths and lower for long wavelengths. As a result, blue light travels more slowly than red light in most media, and Snell’s law predicts a greater angle of refraction for blue light than for

red light (Fig 1-20).

Figure 1-20 Chromatic dispersion. (Illustration developed b y Kevin M. Miller, MD, and rendered b y C.H. Wooley.)

The Abbe number, also known as the V-number, is a measure of a material’s dispersion. Named for the German physicist Ernst Abbe (1840–1905), the Abbe number V is defined as

where nD, nF, and nC are the refractive indices of the Fraunhofer D, F, and C spectral lines (589.2 nm, 486.1 nm, and 656.3 nm, respectively). Low-dispersion materials, which demonstrate low chromatic aberration, have high values of V. High-dispersion materials have low values of V. Abbe numbers for common optical media typically range from 20 to 70.

Reflection and Refraction at Curved Surfaces

For the sake of simplicity, the laws of reflection and refraction were illustrated at flat optical interfaces. However, most optical elements have curved surfaces. To apply the law of reflection or refraction to curved surfaces, the position of the surface normal must be determined because the angles of incidence, reflection, and refraction are defined with respect to the surface normal. Once the position of the surface normal is known, the laws of refraction and reflection define the relationship between the angle of incidence and the angles of refraction and reflection, respectively.

Although there is a mathematical procedure for determining the position of the surface normal in any situation, the details of it are beyond the scope of this text. For selected geometric shapes, however, the position of the surface normal is easy to determine. In particular, the surface normal to a spherical surface always intersects the center of the sphere. For example, Figure 1-21 shows a ray incident on a spherical surface. The incident ray is 2 cm above, and parallel to, the optical axis. The surface normal is found with the extension of a line connecting the center of the sphere to the point where the incident ray strikes the surface. The angle of incidence and the sine of the angle of incidence are determined by simple trigonometry.

Figure 1-21 A ray 2 cm above and parallel to the optical axis is incident on a spherical surface. The surface normal is found by connecting the point where the ray strikes the surface to the center of the sphere (point C). The angle of incidence is found using similar triangles and trigonometry (arctan 2/7 = 16.6°). (Illustration developed b y Edmond H. Thall, MD,

and Kevin M. Miller, MD, and rendered b y C. H. Wooley.)

The Fermat Principle

The mathematician Pierre de Fermat posited that light travels from one point to another along the path requiring the least time. Both Snell’s law of refraction and the law of reflection can be mathematically derived from the Fermat principle. This principle is summarized below and further detailed in Appendix 1.2 at the end of this chapter.

Suppose that the law of refraction were unknown, and consider light traveling from a point source in air, across an optical interface, to some point in glass (Fig 1-22). Unaware of Snell’s law, we might consider various hypothetical paths that light might follow as it moves from point A to point B. Path 3 is a straight line from A to B and is the shortest total distance between the points. However, a large part of path 3 is inside glass, where light travels more slowly. Path 3 is not the fastest route. Path 1 is the longest route from A to B but has the shortest distance in glass. Nevertheless, the extreme length of the overall route makes this a fairly slow path. Path 2 is the best compromise between distance in glass and total path length, and this is the path light will actually follow.

Figure 1-22 Light traveling from points A to B follows only path 2 because it requires the least time. Light does not travel

along either path 1 or path 3. (Illustration developed b y Edmond H. Thall, MD, and Kevin M. Miller, MD, and rendered b y C. H. Wooley.)

Using mathematics beyond the scope of this text, it can be shown that the optimal path is the one predicted by Snell’s law. Thus, Snell’s law is a consequence of the Fermat principle.

Figure 1-23 shows light from an object point traveling along 2 different paths to the image point. According to the Fermat principle, the time required to travel from object to image point (or, alternatively, the optical path length, OPL) must be exactly identical for each path or the paths will not intersect at the image point.