Figure 1-72 Right-angled triangle. (Illustration developed b y Kevin M. Miller, MD, and rendered b y C. H. Wooley.)

The Pythagorean theorem states that c2 = a2 + b2; as a result, therefore, . Triangles are said to be similar when their angles are equal. When 2 triangles have identical angles, their sides are proportional. The triangles in Figure 1-73 are similar.

Figure 1-73 These 2 triangles are similar because their angles are equal. (Illustration developed b y Kevin M. Miller, MD, and

rendered b y C. H. Wooley.)

Appendix 1.2

Light Properties and First-Order Optics

The Fermat principle

As discussed earlier in the chapter, the mathematician Pierre de Fermat believed that natural processes occur in the most economic way. The Fermat principle, as applied to optics, implies that light travels from one point to another along the path requiring the least time. Historically, the laws of reflection and refraction were discovered by careful experimental measurements before Fermat’s time. However, both the law of refraction and the law of reflection can be mathematically derived.

The Fermat principle is an important conceptual and practical tool. The concept of optical path length (OPL) enhances the practical utility of this principle. OPL is the actual distance light travels in a given medium multiplied by the medium’s refractive index. For instance, if light travels 5 cm in air (n = 1.000) and 10 cm in spectacle crown glass (n = 1.523), the OPL is (5 cm × 1.000) + (10 cm × 1.523) = 20.2 cm. According to the Fermat principle, light follows the path of minimum OPL.

Figure 1-23 shows light from an object point traveling along 2 different paths to the image point.

Light traveling path 1 from object to image point traverses a relatively 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. In other words, the time required for light to travel from object to image—and, thus, the OPL—is identical for both paths.

Stigmatic imaging using a single refracting surface

By the early 1600s, the telescope and microscope had been invented. Although the images produced by these early devices were useful, their quality was not very high because the lenses did not focus stigmatically.

At the time, lensmakers were not very particular about the shape of the surfaces that were ground on the lens. It seemed that any curved surface produced an image, so lens surfaces were carefully polished but haphazardly shaped. However, as ideas such as stigmatic imaging and Snell’s law developed, it became clear that the shape of the lens surfaces determined the quality of the image, and so, during the seventeenth century, lensmakers began to carefully shape the lens surface.

The following question arose: what surface produces the best image? Descartes applied the Fermat principle to the simplest situation possible—a single refracting surface. Consider a single object point and a long glass rod (Fig 1-74). Descartes realized that if the end of the rod were configured in a nearly elliptical shape, a stigmatic image would form in the glass. This shape became known as a Cartesian ellipsoid, or Cartesian conoid.

Figure 1-74 The Cartesian conoid is a single refracting surface that produces a stigmatic image for a single object point.

(Illustration developed b y Edmond H. Thall, MD, and Kevin M. Miller, MD, and rendered b y C. H. Wooley.)

Some readers may be troubled by the fact that the image forms in the vitreous cavity, and in an emmetropic eye the image forms on the retina. Once a stigmatic image is produced, the rod is cut and a second Cartesian ellipsoid placed on the back surface (Fig 1-75). The final image is also stigmatic. The Cartesian ellipsoid produces a stigmatic image of only 1 object point. All other object points image nonstigmatically.

Figure 1-75 A combination of Cartesian ellipsoids also gives a stigmatic image. (Illustration developed b y Edmond H. Thall, MD,

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

Until about 1960, it was impossible to manufacture a Cartesian ellipsoid. The only surfaces that could be accurately configured were spheres, cylinders, spherocylinders, and flats. Now, aspheric surfaces are relatively easy to manufacture.

Descartes established that a single refracting surface could, at best, produce a stigmatic image of only 1 object point. By means of mathematics, it has been demonstrated that an optical system can produce a stigmatic image for only as many object points as there are “degrees of freedom” in the optical system. A single lens has 3 degrees of freedom (df): the front surface, the back surface, and the lens thickness. A combination of the 2 lenses has 7 df: the 4 lens surfaces, the 2 lens thicknesses, and the distance between the lenses. Optical systems utilizing multiple lenses improve image quality.

First-order optics

For centuries, the sphere was the only useful lens surface that could be manufactured. Descartes proved that lenses with spherical surfaces do not produce stigmatic images, but common experience shows that such lenses can produce useful images. Consequently, the properties of spherical refracting surfaces have been carefully studied.

The currently accepted approach for studying the imaging properties of any lens is through the method called exact ray tracing. In this technique, Snell’s law is used to trace the paths of several rays, all originating from a single object point. A computer carries out the calculations to as high a degree of accuracy as necessary, usually between 6 and 8 significant figures.

Figure 1-76 shows an exact ray trace for a single spherical refracting surface. Because the image is not stigmatic, the rays do not converge to a single point. However, there is one location where the rays are confined to the smallest area, and this is the location of the image. The distribution of rays at the image location indicates the size of the blur circle, or point spread function (PSF). From the size of the blur circle, the image quality is determined. From the location of the image, other properties, such as magnification, are determined. Ultimately, all image properties may be determined with exact ray tracing.

Figure 1-76 An exact ray trace for a single refracting surface. The image is not stigmatic. However, at one particular location, indicated by the dotted line, the rays are confined to the smallest area. This is the image location. (Illustration

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

Beginning in the 1600s, methods of analyzing optical systems were developed that either greatly reduced or eliminated the need for calculation. These methods are based on approximations—that is, these methods do not give exact answers. Nevertheless, carefully chosen approximations can yield results that are very close to the exact answer while greatly simplifying the mathematics.

The trick is to choose approximations that provide as much simplification as possible while retaining as much accuracy as possible. In this regard, the mathematician Carl Gauss (1777–1855) made many contributions to the analysis of optical systems. Gauss’s work, combined with that of others, developed into a system for analyzing optical systems that has become known as first-order optics.

Ignoring image quality

Determining image quality requires knowledge of how light from a single object point is distributed in the image (ie, the PSF). To determine the PSF, hundreds of rays must be accurately traced. In Gauss’s day, manufacturing techniques rather than optical system design limited image quality. Accordingly, there was little interest in theoretically analyzing image quality. Interest lay instead in analyzing other image features, such as magnification and location.

To determine all image characteristics except image quality requires tracing only a few rays. In fact, if image quality is ignored, analysis of optical systems is reduced from tracing hundreds of rays to tracing just 2 rays. In Gauss’s time, however, tracing even 2 rays exactly was a daunting task, especially if the optical system consisted of several lenses.

Paraxial approximation

To trace a ray through a refracting surface exactly, we need to establish a coordinate system. By convention, the origin of the coordinate system is located at the vertex, the point where the optical axis intersects the surface. Also by convention, the y-axis is vertical, the z-axis coincides with the optical axis, and the x-axis is perpendicular to the page (Fig 1-77). An object point is selected, and a ray is drawn from that object point to the refracting surface.

Figure 1-77 To trace a ray through a refracting surface exactly, it is necessary to establish a coordinate system (the x-, y-, and z-axes) and then find the precise coordinates (y,z) of the point where the ray intersects the surface. (Illustration

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

The first difficulty in making an exact ray trace is determining the precise coordinates (y,z) where the ray strikes the refracting surface. The formula for finding the intersection of a ray with a spherical surface requires fairly complicated calculations involving square roots.

Instead of tracing a ray through an optical system, it is easier to work with rays extremely close to the optical axis, the so-called paraxial rays. The portion of the refracting surface near the optical axis may be treated as flat. Just as the earth’s surface seems flat to a human observer, a refracting surface “seems” flat to a paraxial ray (Fig 1-78). For a ray to be paraxial, it must hug the optical axis over its entire course from object to image. A ray from an object point far off-axis is not paraxial even if it strikes the refracting surface near the axis (Fig 1-79).

Figure 1-78 The paraxial region. The enlargement (bottom) shows the paraxial region with the vertical scale greatly increased but the horizontal dimensions unchanged. Notice that in the paraxial region, the lens is essentially flat. The

paraxial rays are shown in red. (Illustration developed b y Edmond H. Thall, MD, and Kevin M. Miller, MD, and rendered b y C. H. Wooley.)

Figure 1-79 Both rays strike the refracting surface in the paraxial region. Only the lower (red) ray is a paraxial ray, however, because it is close to the optical axis over its entire path. (Illustration developed b y Edmond H. Thall, MD, and Kevin M.

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

Small-angle approximation

To trace a paraxial ray, begin with an object point at or near the optical axis. Then extend a ray from the object point to the refracting surface, which is represented by a flat vertical plane (Fig 1-80). The next step is to determine the direction of the ray after refraction.

Figure 1-80 Detail of the paraxial region with the vertical scale greatly enlarged relative to the horizontal scale. The lens is spherical but appears flat in the paraxial region. The center of the lens is indicated by point C. The points O and I represent the object point and its image, respectively. θi and θt indicate the angles of incidence and transmission, respectively.

(Illustration developed b y Edmond H. Thall, MD, and Kevin M. Miller, MD, and rendered b y C. H. Wooley.)

To determine the direction of the refracted ray, apply Snell’s law. The angle of incidence is θ, and the angle of transmission is θt. Thus,

n sin θ = n′ sin θt

Now the polynomial expansion for the sine function is

where the angle θ is expressed in radians. If the angle θ is small, the third-order term, θ3/3!, and every term after it become insignificant, and the sine function is approximated as

sin θ ≈ θ

This is the mathematical basis of the (essentially equivalent) terms small-angle approximation, paraxial approximation, and first-order approximation. Only the first-order term of the polynomial expansion needs to be used when the analysis is limited to paraxial rays, which have a small angle of entry into the optical system.

The angles appear large in the bottom part of Figure 1-78 because of the expanded vertical scale, but the upper part shows that in the paraxial region these angles are quite small.

Rearranging yields

Finally, we define the refractive power of the surface, P = [(n′ – n)/r]. Thus,

This equation is the lensmaker’s equation. The ratio n/o is the reduced object vergence (U), and the ratio n′/i is the reduced image vergence (V). Vergence is discussed in detail in the section Ophthalmic Lenses.