Light – Electromagnetic energy detectable by the eye.
Geometrical Optics – The properties of light governed by propagation in straight lines, refraction, and reflection.
Physical Optics – The properties of light described by wave phenomena such as interference, diffraction, and polarization.
Quantum Optics – The properties of light described by absorption and emission of energy in discrete quanta, proportional to frequency.
Key Features
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Specular Reflection – Light reflects off smooth surfaces so that the angle of incidence equals the angle of reflection.
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Snell’s Law – The relationship between the bending of light at an interface surface to the speed of light on either side of the interface.
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Vergence Equation – The relationship between the power of a lens and the location of the images it forms.
Introduction
Visible light is the portion of the electromagnetic spectrum that can be detected by the eye. In practice, this ranges from wavelengths of about 750 nm (red) to about 440 nm (violet). Longer wavelengths may be discernable as heat (“infrared”) and can be detected by suitable photographic emulsions and electronic camera chips. Shorter wavelengths (“ultraviolet”) are sometimes visible in eyes after removal of the crystalline lens and can be seen by some insects ( Fig. 2.1.1 ).
The behavior of light under ordinary circumstances is very familiar, but careful observations reveal important subtleties that have fascinated scientists for hundreds of years. In general, the behavior of light in detail depends on the scale of the objects with which it interacts.
Interactions between light and large objects (relative to the wavelength of the light) generally follow simple geometrical rules and come under the heading of “geometrical optics.” This is the regime of typical human experience—light rays travel in straight lines through homogeneous media but may be reflected by polished smooth surfaces or may be refracted (bent) as they pass from one medium to another. These interactions of light with matter are governed by the law of (specular) reflection and Snell’s law, respectively. Geometrical optics is the appropriate tool for understanding the use of lenses for the formation of images—as in the human eye—or as modified by lenses such as spectacles, contact lenses, or intraocular lens implants.
When the dimensions of optical systems are comparable to the wavelength of the light passing through them, the effects of interference become evident, demonstrating the “wave-like” properties of light. Perhaps the most common example is the diffraction of light as it passes through finite apertures, such as the pupil of the human eye. Because the light bends slightly under these circumstances, diffraction limits the sharpness of images formed through small apertures. Wave properties of light also are seen in the phenomena of polarization and are exploited in such optical instruments as interferometers (which accurately measure very small distances) and clinical devices such as the optical coherence tomographer, which exploits interference between beams of light that have scattered from various surfaces within the eye to provide high-resolution images of ocular tissues. These phenomena are discussed later under the heading of “Wave Properties of Light.”
At the smallest scales and energies, the quantum behavior of light becomes evident. Quantum effects are responsible for the operation of lasers, the characteristic absorption and emission spectra of various materials, and the phenomena of fluorescence and phosphorescence.
Geometrical Optics
Under ordinary conditions, light travels through homogeneous media in straight lines. This can be exploited by simple devices such as the “camera obscura” or pinhole camera, which forms images of bright objects by selecting a single ray of light from each point in the source object that threads through a small aperture to form an inverted image on a convenient surface beyond the pinhole. Although these images enjoy an excellent depth of field, bringing objects both near and far into sharp focus, the small aperture limits the amount of light available to form the image ( Fig. 2.1.2 ).
On the other hand, the paths of light rays can be altered by reflection or refraction. In reflection, the incoming ray of light reverses direction to create equal angles between the incoming ray and the exiting ray, as measured from a line through the point of contact perpendicular to the reflecting surface (the “surface normal”). This is the “law of (specular) reflection.” Specular (mirror-like) reflection is seen as light encounters smoothly polished surfaces, such as mirrors, and still pools of liquids, such as water or mercury. Flat reflecting surfaces recreate accurate reproductions of the source objects (with, of course, the left–right direction reversed). Curved reflecting surfaces can be used to magnify or minify the source objects, as used for special purposes such as telescopes, shaving mirrors, or the minifying mirrors rearview mirrors used in automobiles ( Fig. 2.1.3 ).
When light traverses a boundary between two transparent media where the speed of light differs between the two materials, the path of the light may be deflected from a straight line by the process of refraction.
The deviation is described by “Snell’s law” as follows. First, compute the “refractive index” of each material as the ratio of the speed of light in a vacuum divided by the speed of light in the medium—the refractive index is often denoted by the letter n, and is always greater than 1.0 for material media, as light will always travel through a material medium more slowly than through a vacuum. If we measure the angle between the incoming light ray and the surface normal at the point of contact (the “angle of incidence”) and compare it with the angle between the surface normal and the outgoing ray that emerges from the point of contact as the light moves away from the interface in the second medium, we have
n1sinθ1=n2sinθ2
In practice, if the light goes from a “rarer” medium, such as air (with a greater velocity of light and thus a smaller refractive index), to a “denser” medium (with a slower velocity of light and thus a greater refractive index), such as water or glass, the light will bend toward the surface normal. (Of course, this use of the word “dense” has nothing to do with the specific gravity of the materials.) Light that travels from a “slow” medium to a medium with a greater velocity of light will bend away from the surface normal.
The bending of light across such an interface is readily appreciated when looking at objects in a pool of water from the air above, where objects are typically seen as farther away than they really are because the light from the objects bends toward the observer as it passes from the water to the air ( Fig. 2.1.5 ).
Ordinary prisms work the same way. Light passing through a prism is bent toward the base of the prism. Objects viewed through a prism are seen displaced toward the apex. The strength of a prism is usually given in terms of “prism diopters”—a prism that deflects a beam of light by d cm at a distance 1.0 m from the prism is said to have a strength of d prism diopters, usually abbreviated “Δ.”
If the angle of incidence for light going from a “slow” to a “fast” medium exceeds the “critical angle” where sin θ = n 2 / n 1 , then Snell’s law cannot be satisfied, and the light ray is reflected at the interface rather than refracted across it. This “total internal reflection” is employed by prisms in high-quality binoculars, for example ( Fig. 2.1.6 ).
When light passes through a curved surface such as the surface of a lens, the deflection depends in detail on the shape of the surface. For lenses with spherical surfaces, for which one can determine a geometrical center of curvature, and for light rays that closely approximate the line between the source objects and the center of curvature (the “optic axis”), one can use Snell’s law to show that the lens will form a pointlike image of a point source and derive simple rules relating the location of the source objects, the curvature (or power) of the lens, and the location of the image formed by the lens. This is referred to as “stigmatic imagery” ( Fig. 2.1.7 ).
Stigmatic imagery is strictly possible only for “paraxial” rays and lenses with relatively small apertures (though larger than the pinhole apertures described earlier). Nevertheless, this formulation of geometrical optics is very useful in a wide variety of settings, even when the strict assumptions are not met. For example, in the human eye, strict stigmatic imaging is possible only when the pupil is fairly small, but because awareness of the sharpness of the image formed on the retina by the optics of the eye is dominated by the image presented to the fovea, the paraxial regime remains an adequate description and is routinely used to guide the prescription of spectacle lenses and contact lenses.
Basic Stigmatic Optics
A convex lens will image the light from an infinitely distant object (such as a star) at a finite distance, say f, from the lens on the side opposite from the source. This distance is referred to as the focal length of the lens and is measured in meters. The “power” of the lens, P, is given by the equation P = 1/ f. In this context, the units for lens power are referred to as “diopters” and abbreviated “D.” For objects closer than infinity, the image location is determined by the relation
1/u+P=1/v,(*)
For thin lenses, the power of two lenses placed in contact with each other and used as a single lens system is approximately additive: P = P 1 + P 2 .
Lenses can also be fabricated with concave surfaces. These lenses, which are assigned powers less than 0, do not form images by themselves but can be used to adjust the power of convex lenses by means of the addition formula above. Placing such a lens adjacent to an existing lens system can reduce the effective power, for example, pushing an image farther to the right, as described earlier.
If light approaches or leaves a lens through a medium other than air or a vacuum (with n = 1.0), the equation (*) must be modified as follows:
n1/u+P=n2/v
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