Lens (optics)

A lens is a device for either concentrating or diverging light, usually formed from a piece of shaped glass. Analogous devices used with other types of electromagnetic radiation are also called lenses: for instance, a microwave lens can be made from paraffin wax.

In its usual form, a lens consists of a slab of glass or other optically transparent material (such as perspex) with two shaped surfaces of a particular curvature. It is the refractive index of the lens material and the curvature of the two surfaces that give a particular lens its particular properties. A lens works by refracting (bending) the light that passes through it, in a similar manner to a prism.

Table of contents

1 Lens construction 2 Imaging properties 3 Aberrations 3.1 Spherical aberration 3.2 Coma 3.3 Chromatic abberation 4 Multiple lenses 5 Uses of lenses 6 See also 7 External links

Lens construction

The most common type of lenses are spherical lenses, which are formed from surfaces that have spherical curvature, that is, the front and back surfaces of the lens can be imagined to be part of the surface of two spheres of given radii, R₁ and R₂, which are called the radius of curvature of each surface. The sign of R₁ gives the shape of the front surface of the lens: if R₁ is positive, the surface is convex (bulging outwards from the lens). If R₁ is negative, the front surface is concave (bulging into the lens). If R₁ is infinite, the surface is flat, or has zero curvature, and is said to be plane. The same is true for the back surface of the lens, except that the sign conversion is reversed: if R₂ is positive, it is concave, and if R₂ is negative,the back surface is convex. The line joining the centres of the spheres making up the lens surfaces is called the axis of the lens; in almost all cases the lens axis passes through the physical centre of the lens.

Lens are classified by the curvature of these two surfaces. A lens is biconvex if both surfaces are convex, likewise, a lens with two concave surfaces is biconcave. If one of the surfaces is flat, the lens is termed plano-convex or plano-concave depending on the curvature of the other surface. A lens with one convex and one concave side is termed convex-concave, and in this case if both curvatures are equal it is a meniscus lens.

If the lens is biconvex or plano-convex, a collimated or parallel beam of light passing along the lens axis and through the lens will be converged (or focused) to a spot on the axis, at a certain distance behind the lens (known as the focal length). In this case, the lens is called a positive or converging lens.

If the lens is biconcave or plano-concave, a collimated beam of light passing through the lens is diverged (spread); the lens is thus called a negative or diverging lens. The beam after passing through the lens appears to be emanating from a particular point on the axis in front of the lens; the distance from this point to the lens is also known as the focal length, although it is negative with respect to the focal length of a converging lens.

If the lens is convex-concave, whether it is converging or diverging depends on the relative curvatures of the two surfaces. If the curvatures are equal (a meniscus lens), then the beam is neither converged or diverged.

The value of the focal length f for a particular lens can be calculated from the lensmaker's equation:

,

.

Lenses are also reciprocal; i.e. they have the same focal length when light travels from the front to the back as when light goes from the back to the front (although other properties of the lens, such as the aberration [see below] are not necessarily the same in both directions).

Imaging properties

As mentioned above, a positive or converging lens will focus a collimated beam travelling along the lens axis to a spot (known as the focal point) at a distance f from the lens. Conversely, a point source of light placed at the focal point will be converted into a collimated beam by the lens. These two cases are examples of image formation in lenses. In the former case, an object at an infinite distance (as represented by a collimated beam of light) is focused to an image at the focal point of the lens. In the later, an object at the focal length distance from the lens is imaged at infinity. The plane perpendicular to the lens axis situated at a distance f from the lens is called the focal plane.

If the distances from the object to the lens and from the lens to the image are S₁ and S₂ respectively, for a lens of negligible thickness they are found by the thin lens formula:

.

Note that if S₁ < f, S₂ becomes negative, and the image is apparently positioned on the same side of the lens as the object. Although this kind of image, known as a virtual image, cannot be projected on a screen, an observer looking through the lens will see the image in its apparent calculated position. A magnifying glass creates this kind of image.

The magnification of the lens is given by:

,

In the special case that S₁=∞, we have S₂=f and M=-f/∞=0. This corresponds to a collimated beam being focused to a single spot at the focal point. The size of the image in this case is not actually zero, since diffraction effects place a lower limit on the size of the image (see Rayleigh criterion).

The formulas above may also be used for negative (diverging) lens by using a negative focal length (f), but for these lenses only virtual images can be formed.

Aberrations

Lenses do not form perfect images, and there is always some degree of distortion or aberration introduced by the lens which causes the image to be an imperfect replica of the object. Careful design of the lens system for a particular application ensures that the aberration is minimised.

Spherical aberration

Coma

Chromatic abberation

Other kinds of aberration include field curvature, barrel and pincushion distortion, and astigmatism.

Multiple lenses

Lenses may be combined to form more complex optical systems. The simplest case is when lenses are placed in contact: if the lenses of focal lengths f₁ and f₂ are "thin", the combined focal length F of the lenses can be calculated from:

.

Uses of lenses

One important use of lenses is as a prosthetic for the correction of visual impairments such as myopia and farsightedness. See corrective lens, contact lens, eyeglasses.

Another use is in imaging systems such as a monocular, binoculars, telescope, spotting scope, telescopic gun sight, theodolite, microscope, and camera (photographic lens). A single convex lens mounted in a frame with a handle or stand is a magnifying glass.

Radio astronomy and radar systems often use dielectric lenses, commonly called a lens antenna to refract electromagnetic radiation into a collector antenna. The Square Kilometer Array radio telescope will employ such lenses to get a collection area nearly 30 times greater than the, currently, largest antenna ever built.

See also

• Aberration in optical systems
• Photographic lens
• F-number
• Numerical aperture
• Bokeh
• Telescope
• Microscope
• Fresnel lens
• Optical coatings
• Gradient index lens
• History of lensmaking
• Zoom lens
• Gravitational lens
• Dielectric lens

• Lens article at digitalartform.com
• Thin Lens Java applet