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# Lens

In the field of optics, a lens is a device used for focusing or defocusing a beam of light. It is commonly formed from a disk-shaped blank of transparent material, such as glass, plastic, or fused quartz; both sides are ground and polished, with at least one surface being polished with a curve. The word lens is derived from the Latin word for lentil, since the shape of a lens resembles the curved surface of a lentil bean.

Lenses are important in everyday life. Eyes have lenses that can be adjusted by the ciliary muscles surrounding the lens to provide a clear image of objects far away or up close. The ability of the lens to change its focal length diminishes with age, often requiring correction with external lenses (glasses or contact lenses). Lenses are also used in optical instruments such as cameras, telescopes, binoculars, microscopes, and lighthouse assemblies. Lenses come in many differing shapes, with each surface being flat, concave, or convex.

Focusing, or convergence, occurs because the lens refracts light, as is shown in figure 1a: parallel rays enter the convex lens from the left and, since light travels more slowly through the lens than through air, the rays are bent toward the optical axis that runs through the center of the lens. The rays come together at a point in space that is separated from the lens by the focal length (f). Calculating f for a simple spherical lens (which has a curved surface with a spherical shape) is done using the simple formula

where n is the refractive index of the glass, and r1 and r2 are the radii of curvature of the first and second surfaces respectively. The radius value is positive if the surface is concave and negative if convex. From this formula, it is apparent that reducing the radius of curvature of one or both surfaces will shorten the focal length. Flat surfaces have an infinite radius of curvature and therefore do not contribute to focusing. Figure 1b shows the effect of concave surfaces on the focusing of light: the parallel rays entering the lens are bent away from the optical axis and are said to diverge. In this case the lens is called a negative lens. (Note that divergence of the rays is such that they seem to emanate from a point that is one focal length behind the lens.)

Lenses are important because they can be used to form an image of an object. There are two types of images that may be formed. The first is the real image, which is formed on the side of a lens away from the object and can be projected onto viewing screen. The second is a virtual image, which is formed on the object side of the lens and cannot be projected on a screen; however, the virtual image can be viewed by looking into the lens, as with a microscope. The size, position, magnification, and type of image formed by a positive lens depend on the position of the object relative to the focal length. When the object is located two focal lengths away from the lens, the image is real, inverted, and the same size as the object (figure 2a). If an object is moved further away, the image becomes smaller (figure 2b). If the object is moved closer, then the image becomes larger (figure 2c) until the separation is one focal length. If the object is placed less than one focal distance away from the lens, then a virtual, magnified, upright image is formed (figure 2d). Negative lenses always form a virtual image which is smaller than the object (figure 2e). The position of the image is calculated using the simple equation

where o is the distance between the object and the lens and I is the location of the image. A negative value of I indicated that the image is virtual.

Figure 1. Illustration by Hans & Cassidy. Courtesy of Gale Group.

Single lenses may cause several types of aberration, such as chromatic or spherical aberration, which tend to distort an image. For instance, chromatic aberration occurs because the refractive index depends on the wavelength, and so the lens has a different focal length for different wavelengths. Since the human eye detects light over a large range of wavelengths, chromatic aberration causes the colors of the image to separate and blur. This distortion can be corrected using a compound lens (an achromatic pair), in which the chromatic aberration of the first lens is compensated for by the second. A compound lens is usually a pair of lenses glued together in which the two inner surfaces have the same radius of curvature. Spherical aberration, a distortion caused by the spherical shape of the lens, can be reduced by using special combinations of spherical lenses or by using a lens with a different profile. For instance, a lens with a parabolic profile is used instead of a spherical lens to focus a laser beam on a very fine point.

The f-number of a lens is given by the ratio of the focal length of the lens to the aperture, the opening through which light passes. A lens with a large aperture has a small f-number and therefore lets more light through than a smaller diameter lens. Aberrations become increasingly noticeable as the f-number decreases; thus the design of a low f-number lens system is more complex because there are more aberrations in the system that must be reduced. The recent introduction of computers to lens system design has helped produce new systems that perform better than systems designed using the older design techniques.

Figure 2. Illustration by Hans & Cassidy. Courtesy of Gale Group.

Two basic types of lens shapes with light ray projections. A converging lens is thicker in the middle. A diverging lens is thicker at the edges. Illustration by Argosy. The Gale Group.

Many optical instruments require the use of several lenses, firmly held together and relative to one another; such assemblies are called lens systems. The simplest lens system is the telescope, which consists of two lenses, a large diameter objective that gathers as much light as possible, and a smaller eyepiece that aligns the rays of light in order to allow the eye to see the image. The magnification of the telescope is equal to the ratio of the focal lengths of these two lenses. More complex lens systems can be found in the field of photography, either as camera lenses or in enlarging machines. A camera lens is essentially a positive lens that produces a real image at the film plane; however, because of aberrations and the need for different magnifications, most camera lens systems have multiple elements. Wide-angle lenses have an angle-of-view of 90° to 140° (180° for fisheye lenses) and show considerable distortion, particularly around the edges. Standard camera lenses have an angle-of-view of 50° to 60° and telephoto lenses have an angle-of-view of 20° to 40°. Both of these lenses show less distortion. The tele-photo lens is designed to give a long effective focal length (in the range of 85mm-300mm) without the bulk of a long focal length lens; for example, a 200mm tele-photo lens system uses several lenses to produce the 200 mm effect without being four times as long as a 50mm system. Zoom lenses contain elements that can be moved relative to the others in order to change the focal length of the combination, and so a single lens system can take the place of many. However, since the zoom lens system is not optimized for any set focal length, the image is often not as good as that provided by a fixed focal length lens.

The term lens can also be applied to devices that control the divergence of beams other than light beams. For instance, a magnetic lens is used to focus beams of charged particles (such as electrons and protons) and can be found in particle accelerators and television tubes.

## Resources

### Books

Hecht, Jeff. Optics: Light for a New Age. New York: Scribner, 1987.

Kingslake, Rudolph. A History of the Photographic Lens. New York: Academic Press, 1989.

Smithson, Greg. Light and You. Chicago: Physics Press, 1991.

Sobel, Michael. Light. Chicago: University of Chicago Press, 1987.

Iain A. McIntyre

## KEY TERMS

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Aberration

—A distortion or defect in an image formed by a lens.

Magnification

—The ratio between the size of an image seen using an optical instrument and the size of the image seen with the naked eye. A magnification of 10 means that the image is 10 times larger than it would have been if seen unaided.

Optical axis

—An imaginary line running through the center of an optical system, to which all optical elements are aligned.

Parabolic lens

—A lens where the curved surface, or surfaces, can be described by a parabola which is rotated in space. The parabolic lens is used to reduce spherical aberration.

Refraction

—The bending of light that occurs when traveling from one medium to another, such as air to glass or air to water.

Refractive index

—(characteristic of a medium) Degree to which a wave is refracted, or bent.

Spherical lens

—A lens where the curved surface is part of a spherical surface. This is the simplest type of lens to manufacture.