Lens (optics)

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For other uses, see Lens.

A lens.

Lenses can be used to focus light.

A lens is an optical device which transmits and refracts light, converging or diverging the beam.
[citation needed]
A simple lens consists 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. 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.
The variant spelling lense is sometimes seen. While it is listed as an alternative spelling in some
dictionaries, most mainstream dictionaries do not list it as acceptable.[1][2]

Contents
 [hide] 

 1 History
 2 Construction of simple lenses
o 2.1 Types of simple lenses
o 2.2 Lensmaker's equation
 2.2.1 Sign convention of lens radii R1 and R2
 2.2.2 Thin lens equation
 3 Imaging properties
 4 Aberrations
o 4.1 Spherical aberration
o 4.2 Coma
o 4.3 Chromatic aberration
o 4.4 Other types of aberration
o 4.5 Aperture diffraction
 5 Compound lenses
 6 Other types
 7 Uses
 8 See also
 9 References
 10 Bibliography
 11 External links
o 11.1 Simulations

History[edit]
This section requires expansion with: history after 1758. (January 2012)
See also: History of optics and Camera lens
The Nimrud lens

The word lens comes from the Latin name of the lentil, because a double-convex lens is lentil-
shaped. The genus of the lentil plant is Lens, and the most commonly eaten species is Lens
culinaris. The lentil plant also gives its name to a geometric figure.

The oldest lens artifact is the Nimrud lens, dating back 2700 years to ancient Assyria.[3][4] David
Brewster proposed that it may have been used as a magnifying glass, or as a burning-glass to
start fires by concentrating sunlight.[3][5] Another early reference to magnification dates back to
ancient Egyptian hieroglyphs in the 8th century BC, which depict "simple glass meniscal lenses".
[6]

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). Some scholars argue that the archeological evidence indicates that there was widespread
use of lenses in antiquity, spanning several millennia.[7] Such lenses were used by artisans for
fine work, and for authenticating seal impressions. The writings of Pliny the Elder (23–79) show
that burning-glasses were known to the Roman Empire,[8] and mentions what is arguably the
earliest written reference to a corrective lens: Nero was said to watch the gladiatorial games
using an emerald (presumably concave to correct for nearsightedness, though the reference is
vague).[9] Both Pliny and Seneca the Younger (3 BC–65) described the magnifying effect of a
glass globe filled with water.

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 were
capable of concentrating enough sunlight to ignite fires.[10]

Between the 11th and 13th century "reading stones" were invented. Often used by monks to
assist in illuminating manuscripts, these were primitive plano-convex lenses initially made by
cutting a glass sphere in half. As the stones were experimented with, it was slowly understood
that shallower lenses magnified more effectively.

Lenses came into widespread use in Europe with the invention of spectacles, probably in Italy in
the 1280s.[11] This was the start of the optical industry of grinding and polishing lenses for
spectacles, first in Venice and Florence in the thirteenth century,[12] and later in the spectacle-
making centres in both the Netherlands and Germany.[13] Spectacle makers created improved
types of lenses for the correction of vision based more on empirical knowledge gained from
observing the effects of the lenses (probably without the knowledge of the rudimentary optical
theory of the day).[14][15] The practical development and experimentation with lenses led to the
invention of the compound optical microscope around 1595, and the refracting telescope in 1608,
both of which appeared in the spectacle-making centres in the Netherlands.[16][17]

With the invention of the telescope and microscope there was a great deal of experimentation
with lens shapes in the 17th and early 18th centuries trying to correct chromatic errors seen in
lenses. Opticians tried to construct lenses of varying forms of curvature, wrongly assuming
errors arose from defects in the spherical figure of their surfaces.[18] Optical theory on refraction
and experimentation was showing no single-element lens could bring all colours to a focus. This
led to the invention of the compound achromatic lens by Chester Moore Hall in England in 1733,
an invention also claimed by fellow Englishman John Dollond in a 1758 patent.

Construction of simple lenses[edit]

Most lenses are spherical lenses: their two surfaces are parts of the surfaces of spheres, with the
lens axis ideally perpendicular to both surfaces. 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. Typically the lens axis
passes through the physical centre of the lens, because of the way they are manufactured. Lenses
may be cut or ground after manufacturing to give them a different shape or size. The lens axis
may then not pass through the physical centre of the lens.

Toric or sphero-cylindrical lenses have surfaces with two different radii of curvature in two
orthogonal planes. They have a different focal power in different meridians. This is a form of
deliberate astigmatism.

More complex are aspheric lenses. These are lenses where one or both surfaces have a shape that
is neither spherical nor cylindrical. Such lenses can produce images with much less aberration
than standard simple lenses. These in turn evolved into freeform (digital/adaptive/corrected
curve) spectacle lenses, where up to 20,000 ray paths are calculated from the eye to the image
taking into account the position of the eye and the differing back vertex distance of the lens
surface and its pantoscopic tilt and face form angle. The lens surface(s) are digitally adapted at
nanometre levels (normally by a diamond stylus) to eliminate spherical aberration, coma and
oblique astigmatism. This type of lens design almost completely fulfills the sagittal and
tangential image shell requirements first described by Tscherning in 1925 and further described
by Wollaston and Ostwalt.[citation needed] These advanced designs of spectacle lens can improve the
visual performance by up to 70% particularly in the periphery.[citation needed]

Types of simple lenses[edit]

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. If both surfaces have the same radius of
curvature, the lens is equiconvex. 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. It is this type of lens that is most commonly used in corrective lenses.

If the lens is biconvex or plano-convex, a collimated 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.
Convex-concave (meniscus) lenses can be either positive or negative, depending on the relative
curvatures of the two surfaces. A negative meniscus lens has a steeper concave surface and will
be thinner at the centre than at the periphery. Conversely, a positive meniscus lens has a steeper
convex surface and will be thicker at the centre than at the periphery. An ideal thin lens with two
surfaces of equal curvature would have zero optical power, meaning that it would neither
converge nor diverge light. All real lenses have a nonzero thickness, however, which affects the
optical power. To obtain exactly zero optical power, a meniscus lens must have slightly unequal
curvatures to account for the effect of the lens' thickness.

Lensmaker's equation[edit]

The focal length of a lens in air can be calculated from the lensmaker's equation:[19]

where

is the power of the lens,

is the focal length of the lens,
is the refractive index of the lens material,
is the radius of curvature of the lens surface closest to the light source,
is the radius of curvature of the lens surface farthest from the light source, and
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[edit]

Main article: Radius of curvature (optics)

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. With this
convention the signs are determined by the shapes of the lens surfaces, and are independent of
the direction in which light travels through the lens.

Thin lens equation[edit]

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

[20]

The focal length f is positive for converging lenses, and negative for diverging lenses. The
reciprocal of the focal length, 1/f, is the optical power of the lens. If the focal length is in metres,
this gives the optical power in dioptres (inverse metres).

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[edit]

This image has three visible reflections and one visible projection of the same lamp; two
reflections are on a biconvex lens.

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.

The Golden Gate Bridge refracted in rain droplets, which act as lenses

Image of a plant as seen through a biconvex lens.

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
 .

This can also be put into the "Newtonian" form:

[21]

where and .

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 a sharp image of
the object projected onto it, as long as S1 > f (if the lens-to-screen distance S2 is varied slightly,
the image will become less sharp). This is the principle behind photography and the human eye.
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.
Images of black letters in a thin convex lens of focal length f are shown in red. Selected rays are
shown for letters E, I and K in blue, green and orange, respectively. Note that E (at 2f) has an
equal-size, real and inverted image; I (at f) has its image at infinity; and K (at f/2) has a double-
size, virtual and upright image.

Aberrations[edit]

 v
 t
 e

Optical aberration

Distortion

Spherical aberration

Coma
Astigmatism

Petzval field curvature

Chromatic aberration
Defocus
Tilt
Main article: Optical aberration

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[edit]

Main article: 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 distant 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,
but advances in technology have greatly reduced the manufacturing cost for such 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 source.

Coma[edit]

Main article: Coma (optics)

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[edit]

Main article: 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
upon 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.

Different lens materials may also be used to minimise chromatic aberration, such as specialised
coatings or lenses made from the crystal fluorite. This naturally occurring substance has the
highest known Abbe number, indicating that the material has low dispersion.
Other types of aberration[edit]

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

Aperture diffraction[edit]

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[edit]
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 given by

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 in air by some distance d, the focal length for the combined
system is given by

The distance from the second lens to the focal point of the combined lenses is called the back
focal length (BFL).

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 combined focal
length and BFL are infinite. This corresponds to a pair of lenses that transform a parallel
(collimated) beam into another collimated beam. This type of system is called an afocal system,
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 such a 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.

Other types[edit]
Cylindrical lenses have curvature in only one direction. They are used to focus light into a line,
or to convert the elliptical light from a laser diode into a round beam.

Close-up view of a flat Fresnel lens.

A Fresnel lens has its optical surface broken up into narrow rings, allowing the lens to be much
thinner and lighter than conventional lenses. Durable Fresnel lenses can be molded from plastic
and are inexpensive.

Lenticular lenses are arrays of microlenses that are used in lenticular printing to make images
that have an illusion of depth or that change when viewed from different angles.

A gradient index lens has flat optical surfaces, but has a radial or axial variation in index of
refraction that causes light passing through the lens to be focused.

An axicon has a conical optical surface. It images a point source into a line along the optic axis,
or transforms a laser beam into a ring.[22]

Superlenses are made from metamaterials with negative index of refraction. They can achieve
higher resolution than is allowed by conventional optics.

Uses[edit]
A single convex lens mounted in a frame with a handle or stand is a magnifying glass.

Lenses are used as prosthetics 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 usually shaped to fit in a roughly oval, not circular, frame;
the optical centres are placed over the eyeballs; their curvature may not be axially symmetric to
correct for astigmatism. Sunglasses' lenses are designed to attenuate light; sunglass lenses that
also correct visual impairments can be custom made.

Other uses are in imaging systems such as monoculars, binoculars, telescopes, microscopes,
cameras and projectors. 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, or can be viewed on a screen. In these devices lenses are sometimes paired up
with curved mirrors to make a catadioptric system where the lenses spherical aberration corrects
the opposite aberration in the mirror (such as Schmidt and meniscus correctors).

Convex lenses produce an image of an object at infinity at their focus; if the sun is imaged, much
of the visible and infrared light incident on the lens is concentrated into the small image. A large
lens will create enough intensity to burn a flammable object at the focal point. Since ignition can
be achieved even with a poorly made lens, lenses have been used as burning-glasses for at least
2400 years.[23] 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 and 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.

Lenses can become scratched and abraded. Abrasion-resistant coatings are available to help
control this.[24]
 Common Optical Defects in Lens Systems (Aberrations)

Microscopes and other optical instruments are commonly plagued by lens errors that distort the
image by a variety of mechanisms associated with defects (commonly referred to as aberrations)
resulting from the spherical geometry of lens surfaces. There are three primary sources of non-ideal
lens action (errors) that are observed in the microscope.

Of the three major classes of lens errors, two are associated with the orientation of wavefronts and
focal planes with respect to the microscope optical axis. These include on-axis lens errors such as
chromatic and spherical aberration, and the major off-axis errors manifested as coma,
astigmatism, and field curvature. A third class of aberrations, commonly seen in
stereomicroscopes that have zoom lens systems, is geometrical distortion, which includes both
barrel distortion and pincushion distortion.

In general, the ultimate effect of optical aberrations in the microscope is to induce faults in the tiny
features and specimen detail of an image that is being observed or digitally recorded. Lens artifacts
in the microscope were first addressed in the eighteenth century when the London instrument maker
John Dollond discovered that chromatic aberrations could be reduced or eliminated by using a
combination of two different types of glass in the fabrication of lenses. Several decades later, during
the nineteenth century, achromatic (free of chromatic aberration) objectives with a high numerical
aperture were developed, although there were still geometrical distortion problems with the lenses.
Modern glass formulations and antireflective coatings, coupled to advanced grinding and
manufacturing techniques, have all but eliminated a majority of the aberrations from today's
microscope objectives. However, careful attention must still be paid to these artifacts, especially
when conducting high-magnification digital microscopy or when working with stereomicroscopes that
have zoom lens systems.

Chromatic Aberration - One of the most common faults observed in spherical lenses, chromatic
aberration occurs because the lens refracts the various colors present in white light at a different
angle according to the wavelength (see Figure 1). Red light is not refracted at the same angle as
green or blue light so the focal point on the optical axis of the lens is farther away from the lens for
red light. Likewise, green light is focused closer to the lens than red light, and blue light is focused in
a plane that is closest to the lens. This phenomenon is commonly referred to as dispersion and
occurs to a certain degree in all spherically shaped lens elements. The inability of the lens to bring
all of the colors into a common focal plane results in a slightly different image size and focal point for
each of the three predominant wavelength groups. The result is a colored fringe or halo surrounding
the image, with the halo color changing as the focal point of the objective is varied.
 Interactive Java Tutorial

Chromatic Aberration

Explore how the component wavelengths of white light are split

according to frequency when they pass through a simple lens
system to produce the common optical artifact known as chromatic
aberration. 

Chromatic aberration artifacts are compounded by the difference in image magnification that occurs
as a result of the varying focal planes for each color group, an effect termed chromatic difference
of magnification. Aberrations of this type can be significantly reduced, or eliminated, by making
compound lenses that are composed of individual elements having different color-dispersing
properties. A wide variety of optical glasses are now available to lens designers. For example,
crown glass has dispersive properties that enable it to be paired in a lens doublet with a flint glass
element to produce an achromatic doublet lens system that focuses blue and red wavelengths in the
same image plane. Additional refinement of an optical system with even more sophisticated glass
formulas and shapes can reduce chromatic aberration even further.

Spherical Aberration - A potentially serious artifact that can have serious consequences on images
produced by the microscope, spherical aberration is the result of using lenses having spherical
surfaces, which is currently the only practical approach to lens design. Spherical aberration occurs
when light waves passing through the periphery of a lens are not brought into exact focus with those
passing through the center (see Figure 2 for an example using monochromatic red light). The result
is that a well-defined image plane does not exist, and the specimen cannot be correctly focused. As
an example, a point source of light appears as a spot surrounded by a bright halo or series of
diffraction rings when the microscope is brought into its "best" focus. Complex specimens that have
a significant thickness are often so blurred as to be unrecognizable, especially at the periphery of the
viewfield.

Correction of an optical system (such as a microscope) for spherical aberration is often

accomplished by utilizing a combination of positive and negative lens elements with different
thickness, which are cemented together to form a compound lens group. Spherical aberrations are
very important in terms of the resolution of a lens because they affect the coincident imaging of
points along the optical axis and degrade the performance of the lens, which will seriously affect
specimen sharpness and clarity. These lens defects can frequently be reduced by limiting the outer
edges of the lens from exposure to light using diaphragms, and also by utilizing aspherical lens
surfaces within the optical system.

The highest-quality modern microscope objectives address spherical aberrations in a number of

ways including special lens-grinding techniques, improved glass formulations, and better control of
optical pathways. Objectives that are highly corrected for spherical aberration are often designed for
specific conditions, such as strict cover glass thickness restrictions, oil immersion, and a narrow
refractive index tolerance. Adjustable correction collars are available on some high-dry oil-free
objectives to account for cover glass thickness variations. Microscope operators should carefully
study the basic requirements of specialized objectives to make certain spherical aberration is not
introduced due to utilization of the objective under conditions for which it was not designed.
Coma - Similar to spherical aberration in many respects, coma is generally encountered with off-axis
light rays and is most severe when the microscope is out of proper alignment. The aberration is
named for its strong resemblance to the shape of a comet tail, and is manifested by a streak of light
that appears to emanate from a focused spot at the periphery of the viewfield. Coma is often
considered the most problematic aberration due to the asymmetry it produces in images. It is also
one of the easiest aberrations to demonstrate. For example, on a bright, sunny day, when a
magnifying glass is used to focus an image of the sun on the sidewalk, coma aberration can be seen
in the image when the magnifying glass is tilted with respect to the principal rays from the sun. The
sun's image, when projected onto the concrete, will elongate into a comet-like shape that is
characteristic of coma aberration.

Interactive Java Tutorial

 Coma Aberration

Discover how the off-axis coma aberration affects digital images

captured in the microscope. The tutorial explores progressive
degrees of coma and demonstrates the detrimental effect on
resulting images. 

The distinct shape displayed by images suffering from coma aberration is the result of refraction
differences by light rays passing through the various lens zones as the incident angle becomes more
oblique (off-axis). The severity of comatic aberration is a function of thin lens shape. In the extreme,
coma results in meridional rays passing through the periphery of the lens to arrive at the image
plane closer to the axis than do light rays passing through the central portion of the lens (and closer
to the principal ray, as illustrated in Figure 3). In this case, the peripheral rays produce the smallest
image and the coma aberration is said to be negative. In contrast, when the peripheral rays are
focused farther down the axis to produce a much larger image, the aberration is termed positive.
The "comet" shape may have its "tail" pointing toward the center of the viewfield or away, depending
upon whether the aberration has a negative or positive value, respectively. The degree of coma
aberration is greater for lenses with wider apertures, and can be corrected (in part) by reducing
aperture size. Microscope designers usually attempt to correct coma aberration to accommodate the
diameter of the object field for a given objective and eyepiece combination.

Astigmatism - Astigmatism aberration is similar to coma; however, this artifact is not as sensitive to
aperture size and depends more strongly on the oblique angle of the light beam. The aberration is
manifested by the off-axis image of a specimen point appearing as a line or ellipse instead of a
discrete point. Depending on the angle of the off-axis light rays entering the lens, the line image may
be oriented in either of two different directions (see Figure 4), tangentially (meridionally) or sagitally
(equatorially). The intensity ratio of the unit image will diminish, with definition, detail, and contrast
being lost as the distance from the center is increased.

In less expensive microscopes, astigmatism is often the result of asymmetric lens curvature due to
mistakes in manufacture or improper mounting of a lens in its frame or orientation within the
objective barrel. Astigmatism lens errors are usually corrected by designing microscope objectives to
provide precise spacing of individual lens elements as well as appropriate lens shapes and refractive
indices. Careful alignment and adjustment of the individual lens elements is accomplished with
spacers and shims to reduce or eliminate the effects of astigmatism.

Field Curvature - Also commonly referred to as curvature of field, this aberration, which is the
natural result of employing lenses that have curved surfaces, is very familiar to many experienced
microscopists. When light is focused through a curved lens, the image plane produced by that lens
will be curved, as illustrated in Figure 5. The image can be focused over the range between points A
and B to produce a sharp focus either on the edges or in the center. Categorized as an off-axis
aberration, field curvature produces an image plane having the shape of a concave spherical surface
(resembling a convex lens surface), as seen from the objective. Although successive zones can be
brought into focus by translating the objective, the entire image cannot be simultaneously focused
onto a flat surface such as a film plane or the surface of a CCD or CMOS image sensor.

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Geometrical Distortion

Examine how distortion aberrations are manifested in changes to

image shape rather than a reduction in sharpness or alterations to
color balance. 

Optical designers deal with field curvature by adding corrective lens elements to the objective in
specially designed flat-field objectives. Although the optical correction for field curvature requires
the addition of several new lenses to the design, these objectives (termed plan or plano) are the
most common type of objective in use today. Field curvature is seldom totally eliminated, but it is
often difficult to detect edge curvature with most plan-corrected objectives. As a result, very limited
degrees of field curvature often do not degrade photomicrographs or digital images. The artifact is
more severe at low magnifications and can be a serious problem with stereomicroscopes.

Geometrical Distortion - Image distortion is an aberration commonly observed in

stereomicroscopy, and is manifested by changes in the shape of an image rather than the sharpness
or color spectrum. The two most prevalent types of geometrical distortion, positive and negative
(often termed pincushion and barrel, respectively), can often be present in very sharp images that
are otherwise well-corrected for spherical and chromatic aberrations, as well as coma and
astigmatism. When images suffer from distortion, the true geometry of a specimen is no longer
maintained in the image. Figure 6 illustrates examples of rather significant pincushion and barrel
distortion in the image of a computer microprocessor integrated circuit.

Geometric distortion can be difficult to detect, especially when the aberration is relatively slight and
the specimen lacks periodic structures. This type of artifact is most severe in specimens that have
straight lines, such as periodic grids, squares, rectangles or other regular polygonal features that
readily show the curvature present from distortion. Distortion is often found in optical designs utilizing
compound lens systems (telephoto, fisheye, and zoom) containing meniscus, concave,
hemispherical, and thick convex lenses. Complex lens systems, such as the zoom design, can have
rather pronounced distortion, which may vary with focal length, producing pincushion distortion at
long focal lengths and barrel distortion at short focal lengths. For this reason, stereoscopic zoom
microscopes classically have a significant amount of distortion present and microscope
manufacturers have expended considerable effort in alleviating this aberration.