# Lens (optics)

*This is about the optical device. For other uses, see the list at lens.*

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 |

2 Imaging properties 3 Aberrations 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, *R*_{1} and *R*_{2}, which are called the *radius of curvature* of each surface. The sign of *R*_{1} gives the shape of the front surface of the lens: if *R*_{1} is positive, the surface is *convex* (bulging outwards from the lens). If *R*_{1} is negative, the front surface is *concave* (bulging into the lens). If *R*_{1} 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 *R*_{2} is positive, it is concave, and if *R*_{2} 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*:

- ,

*n*is the refractive index of the lens material and

*d*is the distance along the lens axis between the two surfaces (known as the thickness of the lens). If

*d*is small compared to

*R*

_{1}and

*R*

_{2}, then the

*thin lens*assumption can be made, and

*f*can be estimated as:

- .

*f*is positive for converging lenses, negative for diverging lenses, and infinite for meniscus lenses. The value 1/

*f*is known as the

*power*of the lens, and so meniscus lenses are said to have zero power. Lens power is measured in

*dioptres*, which have units of inverse meters (m

^{-1}).

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

What this means is that, if an object is placed at a distance*S*

_{1}along the axis in front of a positive lens of focal length

*f*, a screen placed at a distance

*S*

_{2}behind the lens will have an image of the object projected onto it, as long as

*S*

_{1}>

*f*. This is the principle behind photography. The image in this case is known as a

*real image*.

Note that if *S*_{1} < *f*, *S*_{2} 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:

- ,

*M*is the magnification factor; if |

*M*|>1, the image is larger than the object. Notice the sign convention here shows that, if

*M*is negative, as it is for real images, the image is upside-down with respect to the object. For virtual images,

*M*is positive and the image is upright.

In the special case that *S*_{1}=∞, we have *S*_{2}=*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

### Spherical aberration

There are several different types of aberration which can affect image quality.*Spherical aberration*is caused because spherical surfaces are not the ideal shape with which to make a lens, but they are by far the simplest shape to which glass can be ground and polished and so are often used. Spherical aberration causes beams parallel to but away from the lens axis to be focused in a slightly different place than beams close to the axis. This manifests itself as a blurring of the image. Lenses in which closer-to-ideal, non-spherical surfaces are used are called

*aspheric*lenses, which are complex to make and often extremely expensive. Spherical aberration can be minimised by careful choice of the curvature of the surfaces for a particular application: for instance, a plano-convex lens which is used to focus a collimated beam produces a sharper focal spot when used with the convex side towards the beam.

### Coma

Another type of aberration is*coma*, which derives its name from the comet-like appearance of the aberrated image. Coma occurs when an object off the optical axis of the lens is imaged, where rays pass through the lens at an angle to the axis θ. Rays which pass through the centre of the lens of focal length

*f*are focussed at a point with distance

*f*tan θ from the axis. Rays passing through the outer margins of the lens are focussed at different points, either further from the axis (positive coma) or closer to the axis (negative coma). In general, a bundle of parallel rays passing through the lens at a fixed distance from the centre of the lens are focussed to a ring-shaped image in the focal plane, known as a

*comatic circle*. The sum of all these circles results in a V-shaped or comet-like flare. As with spherical aberration, coma can be minimised (and in some cases eliminated) by choosing the curvature of the two lens surfaces to match the application. Lenses in which both spherical aberration and coma are minimised are called

*bestform*lenses.

### Chromatic abberation

*Chromatic aberration*is caused by the dispersion of the lens material, the variation of its refractive index

*n*with the wavelength of light. Since from the formulae above

*f*is dependent on

*n*, if follows that different wavelengths of light will be focused to different positions. Chromatic aberration of a lens is seen as fringes of color around the image. It can be minimised by using an

*achromatic doublet*(or

*achromat*) in which two materials with differing dispersion are bonded together to form a single lens. This reduces the amount of chromatic aberration over a certain range of wavelengths, though it does not produce perfect correction. The use of achromats was an important step in the development of the optical microscope.

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

## Multiple lenses

Since 1/*f*is the power of a lens, it can be seen that the powers of thin lenses in contact are additive.

## Uses of lenses

One important use of lenses is as a prosthetic for the correction of visual impairments such as myopia, hyperopia, presbyopia, and astigmatism. 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