This article is about the optical device. For other uses, see
lens.
A lens (or lense) is an optical device with perfect or approximate
axial symmetry which transmits and
refracts light, concentrating or diverging the beam. A
simple lens is a lens consisting of a single optical element. A compound lens is an
array of simple lenses (elements) with a common axis; the use of multiple elements allows more optical aberrations to be corrected than is possible with a single element. Manufactured
lenses are typically made of glass or transparent
plastic. Elements which refract electromagnetic
radiation outside the visual spectrum are also called lenses: for instance, a
microwave lens can be made from paraffin wax.
History
- See also: History of optics
The oldest lens artefact is dated to c.640 BC, a rock crystal
lens found at excavations in Niniveh. The earliest written records of lenses date to
Ancient Greece, with Aristophanes' play
The Clouds (424 BC) mentioning a burning-glass (a biconvex lens used to focus the sun's rays to produce fire). The writings of Pliny the Elder (23–79) also show that
burning-glasses were known to the Roman Empire[1], and mentions what is possibly the first use of a corrective lens: Nero was said to watch the gladiatorial games using an emerald[2] (presumably concave to correct for myopia, though
the reference is vague). Both Pliny and Seneca the Younger (3
BC–65) described the magnifying effect of a glass globe filled with water.
The Arabian mathematician Ibn Sahl
(c.940–c.1000) used what is now known as Snell's law to calculate the shape of lenses.[3] Ibn al-Haitham (965–1038) wrote the first major optical treatise, the
Book of Optics, which described how the lens in the human eye formed an image on the retina.
Excavations at the Viking harbour town of Fröjel,
Gotland, Sweden discovered in 1999 the rock crystal
Visby lenses, produced by turning on pole-lathes at Fröjel in the 11th to 12th century,
with an imaging quality comparable to that of 1950s aspheric lenses. The Viking lenses concentrate sunlight enough to ignite
fires.
Widespread use of lenses did not occur until the use of reading stones in the 11th
century and the invention of spectacles, probably in Italy in the
1280s. Nicholas of Cusa is believed to have been the
first to discover the benefits of concave lenses for the treatment of myopia in 1451.
The Abbe sine condition, due to Ernst
Abbe (1860s), is a condition that must be fulfilled by a lens or other optical system in order for it to produce sharp
images of off-axis as well as on-axis objects. It revolutionized the design of optical instruments such as microscopes, and helped to establish the Carl Zeiss company as a leading
supplier of optical instruments.
Construction of simple lenses
Image of the city of
Seattle as seen through a lens.
Most lenses are spherical lenses: their two surfaces are parts, with the same axis as each other, of the surfaces of
spheres. Each surface can be convex
(bulging outwards from the lens), concave
(depressed into the lens), or planar (flat). 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.
Types of simple lenses
Lenses are classified by the curvature of the two optical surfaces. A lens is biconvex (or double convex, or
just convex) if both surfaces are convex, A lens with two concave surfaces is biconcave (or just concave).
If one of the surfaces is flat, the lens is plano-convex or plano-concave depending on the curvature of the other
surface. A lens with one convex and one concave side is convex-concave or meniscus.
If the lens is biconvex or plano-convex, a collimated or parallel beam of light
travelling parallel to the lens axis and passing 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 (a meniscus lens), whether it is converging or diverging depends on the relative curvatures of
the two surfaces. If the curvatures are equal, then the beam is neither converged nor diverged.
Lensmaker's equation
The focal length of a lens in air can be calculated from the lensmaker's equation:[4]
![\frac{1}{f} = (n-1) \left[ \frac{1}{R_1} - \frac{1}{R_2} + \frac{(n-1)d}{n R_1 R_2} \right],](http://content.answers.com/main/content/wp/en/math/d/5/d/d5d9b5b95cf3fd9ea7f2549ae4ee8fa6.png)
where
- f is the focal length of the lens,
- n is the refractive index of the lens
material,
- R1 is the radius of curvature of the lens surface closest to the light
source,
- R2 is the radius of curvature of the lens surface farthest from the light
source, and
- d is the thickness of the lens (the distance along the lens axis between the two
surface vertices).
Sign convention of lens radii R1 and R2
-
The signs of the lens' radii of curvature indicate whether the corresponding surfaces are convex or concave. The
sign convention used to represent this varies, but in this article if
R1 is positive the first surface is convex, and if R1 is negative the surface is concave. The
signs are reversed for the back surface of the lens: if R2 is positive the surface is concave, and if
R2 is negative the surface is convex. If either radius is infinite, the
corresponding surface is flat.
Thin lens equation
If d is small compared to R1 and R2, then the thin
lens approximation can be made. For a lens in air, f is then given by
![\frac{1}{f} \approx \left(n-1\right)\left[ \frac{1}{R_1} - \frac{1}{R_2} \right].](http://content.answers.com/main/content/wp/en/math/a/f/a/afad8f375f5846cea8f9e8091f145d60.png)
[5]
The focal length f is positive for converging lenses, negative for diverging lenses, and infinite for meniscus lenses.
The value 1/f is known as the optical power of the lens, and so meniscus
lenses are said to have zero power. Lens power is measured in dioptres, which are units
equal to inverse meters (m−1).
Lenses have the same focal length when light travels from the back to the front as when light goes from the front to the back,
although other properties of the lens, such as the aberrations are not
necessarily the same in both directions.
Imaging properties
As mentioned above, a positive or converging lens in air 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 waves) is focused to an image at the focal point
of the lens. In the latter, 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, in air, the distances are related by the thin lens formula:
.
What this means is that, if an object is placed at a distance S1 along the axis in front of a positive lens
of focal length f, a screen placed at a distance S2 behind the lens will have an image of the object
projected onto it, as long as S1 > f. This is the principle behind photography. The image in this case is known as a real image.
Note that if S1 < f, S2 becomes negative, 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:
,
where 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 S1 = ∞, then 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.
For the case of lenses that are not thin, or for more complicated multi-lens optical systems, the same formulas can be used,
but S1 and S2 are interpreted differently. If the system is in air or vacuum, S1 and S2 are measured from the front and rear principal planes of the system, respectively. Imaging in media with an index of refraction
greater than 1 is more complicated, and is beyond the scope of this article.
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 minimized. There are several different types of aberration which can affect image quality.
Spherical aberration
Spherical aberration occurs 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. These were formerly complex to make
and often extremely expensive, although advances in technology have greatly reduced the cost of manufacture for these lenses.
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 focused at a point with distance f tan θ from the axis. Rays passing
through the outer margins of the lens are focused 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 focused 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 aberration
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, it follows
that different wavelengths of light will be focused to different positions. Chromatic aberration of a lens is seen as fringes of
colour 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. An apochromat is a lens or lens system which has even better correction of chromatic aberration, combined
with improved correction of spherical aberration. Apochromats are much more expensive than achromats.
Other kinds of aberration include field curvature,
barrel and pincushion
distortion, and astigmatism.
Aperture diffraction
Even if a lens is designed to minimize or eliminate the aberrations described above, the image quality is still limited by the
diffraction of light passing through the lens' finite aperture. A diffraction-limited
lens is one in which aberrations have been reduced to the point where the image quality is primarily limited by diffraction under
the design conditions.
Compound lenses
- See also: Photographic lens, Doublet (lens), and Achromat
Simple lenses are subject to the optical aberrations discussed above. In many cases these
aberrations can be compensated for to a great extent by using a combination of simple lenses with complementary aberrations. A
compound lens is a collection of simple lenses of different shapes and made of materials of different refractive indices,
arranged one after the other with a common axis.
The simplest case is where lenses are placed in contact: if the lenses of focal lengths f1 and
f2 are "thin", the combined focal length f of the lenses is:
.
Since 1/f is the power of a lens, it can be seen that the powers of thin lenses in contact are additive.
If two thin lenses are separated by some distance d, the distance from the second lens to the focal point of the
combined lenses is called the back focal length (BFL). This is given by:
.
Note that as d tends to zero, the value of the BFL tends to the value of f given for thin lenses in contact.
If the separation distance is equal to the sum of the focal lengths (d = f1+f2),
the BFL is infinite. This corresponds to a pair of lenses that transform a parallel (collimated) beam into another collimated
beam. This type of system is called afocal, since it produces no net convergence or divergence of the beam. Two lenses at
this separation form the simplest type of optical telescope.
Although the system does not alter the divergence of a collimated beam, it does alter the width of the beam. The magnification
of the telescope is given by:
,
which is the ratio of the input beam width to the output beam width. Note the sign convention: a telescope with two convex
lenses (f1 > 0, f2 > 0) produces a negative magnification, indicating an inverted
image. A convex plus a concave lens (f1 > 0 > f2) produces a positive magnification and
the image is upright.
Uses of lenses
A single convex lens mounted in a frame with a handle or stand is a magnifying
glass.
Lenses are used as prosthetic for the correction of visual impairments such as myopia, hyperopia, presbyopia, and astigmatism. See corrective lens, contact lens, eyeglasses. Most lenses used for other purposes have strict
axial symmetry; eyeglass lenses are only approximately symmetric. They are shaped to fit
in a usually roughly oval, not circular, frame; the optical centers are placed over the eyeballs;
their curvature may not be axially symmetric to correct for astigmatism. Sunglasses lenses may be designed to attenuate light without refraction.
Another use is in imaging systems such as a monocular, binoculars, telescope, spotting
scope, telescopic gun sight, theodolite,
microscope, camera (photographic lens) and projector. Some of these instruments produce
a virtual image when applied to the human eye; others produce a real image which can be captured on photographic film or an
optical sensor.
Convex lenses produce an image of an object at infinity at their focus; if the sun is imaged, all
the infrared energy incident on the lens is concentrated on the small image. A large lens will concentrate enough energy to heat
an inflammable object on which the image falls to burning point. Such lenses, which do not need to be even approximately
optically accurate, have been used as burning-glasses for hundreds of years. A modern
application is the use of relatively large lenses to concentrate solar energy on relatively small photovoltaic cells, harvesting more energy without the need to use larger, more expensive, cells.
Radio astronomy and radar systems often use
dielectric lenses, commonly called a lens antenna to refract
electromagnetic radiation into a collector antenna. The Square Kilometre Array radio telescope, scheduled to be
operational by 2020[1], will employ such lenses to get a collection area nearly 30 times greater than any previous antenna.
See also
References
General
- Hecht, Eugene (1987). Optics, 2nd ed., Addison
Wesley. ISBN 0-201-11609-X.
Chapters 5 & 6.
- Greivenkamp, John E. (2004). Field Guide to
Geometrical Optics, SPIE Field Guides vol. FG01, SPIE. ISBN 0-8194-5294-7.
Footnotes
- ^ Pliny the Elder, The Natural
History (trans. John Bostock) Book XXXVII, Chap. 10.
- ^ Pliny the Elder, The Natural History (trans. John Bostock) Book XXXVII, Chap.
16
- ^ Rashed, R. (1990). "A pioneer in anaclastics: Ibn Sahl on burning mirrors
and lenses." Isis, 81, 464–491.
- ^ Greivenkamp, p.14; Hecht §6.1
- ^ Hecht, § 5.2.3
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