Figure 1-12 A, Textbooks often illustrate images produced by lenses as stigmatic. B, In most cases, however, the images are not stigmatic. The point spread function reveals how faithfully an imaging system reproduces each object point.
(Illustration developed b y Kevin M. Miller, MD, and rendered b y C. H. Wooley.)
To summarize, a stigmatic image is a perfect point image of an object point. However, in most cases, images are not stigmatic. Instead, light from a single object point is distributed over a small region of the image known as a blur circle or, more generally, a PSF. The image formed by an optical system is the spatial summation of the PSF for every object point. The amount of detail in an image is related to the size of the blur circle or PSF for each object point. The smaller the PSF, the better the resemblance between object and image.
Light Propagation
An intensive investigation of light propagation was begun in the late 1500s. Numerous experiments measuring light deviation were conducted, and the data were collected and summarized as laws. These laws are described in the following sections.
Optical Media and Refractive Index
Light travels through a variety of materials, such as air, glass, plastics, liquids, crystals, some biological tissues, the vacuum of space, and even some metals. A medium is any material that transmits light.
Light travels at different speeds in different media. Light moves fastest in a vacuum and slower through any material. The refractive index of an optical medium is the ratio of the speed of light in a vacuum to the speed of light in the medium and is usually denoted in mathematical equations by the lowercase letter n. The speed of light in a vacuum is 299,792,458 m/s. This speed is approximately equal to 300 million meters per second, or 186,000 miles per second. In 1983, the Système International defined a meter as the distance light travels in a vacuum during 1/299,792,458 of a second. (This concept is discussed in greater detail in Chapter 8.) Refractive index is always greater than or equal to 1. In computations, it is often easier to work with the refractive index of a material than directly with the speed of light.
The refractive index,
is quite sensitive to a material’s chemical composition. A small amount of salt or sugar dissolved in water changes its refractive index. Because refractive index is easy to measure accurately, chemists use it to identify compounds or determine their purity. Glass manufacturers alter the refractive index of glass by adding small amounts of rare earth elements. Until recently, clinical laboratories screened for diabetes mellitus by measuring the refractive index of urine. Table 1-1 lists the refractive indices of various tissues and materials of clinical interest.
Table 1-1
Refractive index varies with temperature and barometric pressure, but these changes are usually small enough to be ignored. One exception is for silicone polymer. The refractive index of polymerized silicone at room temperature (20°C) differs enough from its index at eye temperature (35°C) that manufacturers of silicone intraocular lenses (IOLs) have to account for the variation.
Refractive index also varies with wavelength. As discussed in Chapter 8, physical optics regards light in the spectrum of electromagnetic waves. The visual system perceives different wavelengths of light as different colors. Long wavelengths appear red, intermediate wavelengths appear yellow or green, and short wavelengths appear blue. In a vacuum, all wavelengths travel at the same speed. In other mediums, short wavelengths usually travel more slowly than long wavelengths. This phenomenon is called dispersion.
In the human eye, chromatic dispersion leads to chromatic aberration. If yellow wavelengths are focused precisely on the retina, blue light will be focused in front of the retina and red light will be focused behind the retina. (See Clinical Example 1-3.)
Some media, such as quartz, are optically inhomogeneous. That is, the speed of light through the material depends on the direction of light propagation through the material.
Figure 1-13 Chromostereopsis is demonstrated by this illustration of red and blue print on a black background. The illustration is not very dramatic unless rendered on a computer monitor or projected onto a screen. (Illustration developed b y
Kevin M. Miller, MD, and rendered b y C. H. Wooley.)
Clinical Example 1-3
You may notice that red objects appear nearer than blue objects when they are displayed against a black background (Fig 1-13). This effect stands out in slide presentations that are rich in red and blue text and is known as chromostereopsis. It occurs because the human eye has approximately 0.5 D of chromatic aberration. Even individuals with red-green color blindness can observe the effect. To bring red print into focus, the eye must accommodate. To bring blue print into focus, the eye must relax accommodation. As a result, red print appears closer than blue print. The accommodative effort required to bring the various pieces of a chromatic image into focus imparts a 3-dimensional quality to the image.
Law of Rectilinear Propagation
The law of rectilinear propagation states that light in a homogeneous medium travels along straightline paths called rays. The light ray is the most fundamental construct in geometric optics. Of particular note, rays traversing an aperture continue in straight lines in geometric optics. As stated earlier, a bundle of light rays traveling close to each other in the same direction is known as a pencil of light.
The law of rectilinear propagation is inaccurate insofar as it does not account for the effect of diffraction as light traverses an aperture (see Chapter 8). The basic distinction between physical and geometric optics is that geometric optics ignores diffraction because it is based on the law of rectilinear propagation. For clinical purposes, diffraction effects are rarely important. However, in situations for which diffraction effects are significant, geometric optics does not fully describe the image.
Optical Interfaces
The boundary between 2 different optical media is called an optical interface. Typically, when light reaches an optical interface, some light is transmitted through the interface, some is reflected, and some is absorbed, or converted to heat, by the interface. The amount of light transmitted, reflected, and absorbed depends on several factors.
Law of Reflection (Specular Reflection)
In specular reflection, the direction of the reflected ray bears a definite relationship to the direction of the incident ray. To express a precise relationship between incident rays and reflected rays, it is necessary to construct an imaginary line perpendicular to the optical interface at the point where the incident ray meets the interface. This imaginary line is a surface normal (Fig 1-14). The surface normal and the incident ray together define an imaginary plane known as the plane of incidence and reflection. The angle formed by the incident ray and surface normal is the angle of incidence, θi. This is not the angle between the incident ray and the optical interface. The reflected ray and the surface normal form the angle of reflection, θr.
Figure 1-14 The law of specular reflection. The angle of reflection (θr) is equal to the angle of incidence (θi) and lies in the
same plane (in this case the plane of the paper) that contains the incident ray and the “normal” perpendicular to the surface.
The law of reflection states that the reflected ray lies in the same plane as the incident ray and the surface normal (ie, the reflected ray lies in the plane of incidence) and that θi = θr.
The amount of light reflected from a surface depends on θi and the plane of polarization of the light. The general expression for reflectivity is derived from the Fresnel equations, which are beyond the scope of this text. The reflectivity at normal incidence is simple and depends only on the optical media bounding the interface. The reflection coefficient for normal incidence is given by
The reflection coefficient is used to calculate the amount of light transmitted at an optical interface if absorption losses are minimal.
Law of Refraction (Specular Transmission)
In specular transmission, the transmitted ray’s direction bears a definite relation to the incident ray’s direction. Again, a surface normal is constructed, and the angle of incidence and the plane of incidence and transmission are defined just as they were for reflection (Fig 1-15). The angle formed by the transmitted ray and the surface normal is the angle of refraction, also known as the angle of
transmission. The angle of transmission, θt, is preferred by some authors because the symbol for angle of refraction, θr, might otherwise be confused with that of the angle of reflection, θr.
Figure 1-15 Light moving from a lower index to a higher one bends toward the surface normal (A), and that from a higher
to a lower index bends away from the surface normal (B). (Illustration developed b y Edmond H. Thall, MD, and Kevin M. Miller, MD, and rendered b y C. H. Wooley.)
At the optical interface, light undergoes an abrupt change in speed that, in turn, usually produces an abrupt change in direction. The law of refraction, also known as Snell’s law in honor of its discoverer, states that the refracted, or transmitted, ray lies in the same plane as the incident ray and the surface normal and that
ni sin θi = nt sin θt
where
ni = refractive index of incident medium θi = angle of incidence
nt = refractive index of transmitted medium θt = angle of transmission (or refraction)
When light travels from a medium of lower refractive index to a medium of higher refractive index, it bends toward the surface normal. Conversely, when light travels from a higher to a lower refractive index, it bends away from the surface normal (Clinical Example 1-4; see Fig 1-15).
Clinical Example 1-4
Imagine you are fishing from a pier and you spot a “big one” in front of you a short distance below the surface of the water. You don’t have a fishing rod, but instead you are armed with a spear (Fig 1-16). How should you throw the spear to hit the fish?