Magnification

Magnification is the process of enlarging the apparent size, not

physical size, of something. This enlargement is quantified by a
size ratio called optical magnification. When this number is less
than one, it refers to a reduction in size, sometimes called de-
magnification.

Typically, magnification is related to scaling up visuals or images

to be able to see more detail, increasing resolution, using
microscope, printing techniques, or digital processing. In all cases, The postage stamp appears larger
with the use of a magnifying glass.
the magnification of the image does not change the perspective of
the image.

Examples of magnification
Some optical instruments provide visual aid by magnifying small
or distant subjects.

A magnifying glass, which uses a positive (convex) lens

to make things look bigger by allowing the user to hold
them closer to their eye. Stepwise magnification by 6% per
frame into a 39-megapixel image. In
A telescope, which uses its large objective lens or
the final frame, at about 170x, an
primary mirror to create an image of a distant object and
image of a bystander is seen
then allows the user to examine the image closely with a
smaller eyepiece lens, thus making the object look reflected in the man's cornea.
larger.
A microscope, which makes a small object appear as a
much larger image at a comfortable distance for viewing. A microscope is similar in layout to
a telescope except that the object being viewed is close to the objective, which is usually
much smaller than the eyepiece.
A slide projector, which projects a large image of a small slide on a screen. A photographic
enlarger is similar.
A zoom lens, a system of camera lens elements for which the focal length and angle of view
can be varied.

Size ratio (optical magnification)

Optical magnification is the ratio between the apparent size of an object (or its size in an image) and its
true size, and thus it is a dimensionless number. Optical magnification is sometimes referred to as
"power" (for example "10× power"), although this can lead to confusion with optical power.

Linear or transverse magnification

For real images, such as images projected on a screen, size means a linear dimension (measured, for
example, in millimeters or inches).

Angular magnification
For optical instruments with an eyepiece, the linear dimension of the image seen in the eyepiece (virtual
image at infinite distance) cannot be given, thus size means the angle subtended by the object at the focal
point (angular size). Strictly speaking, one should take the tangent of that angle (in practice, this makes a
difference only if the angle is larger than a few degrees). Thus, angular magnification is given by:

where is the angle subtended by the object at the front focal point of the objective and is the angle
subtended by the image at the rear focal point of the eyepiece.

For example, the mean angular size of the Moon's disk as viewed from Earth's surface is about 0.52°.
Thus, through binoculars with 10× magnification, the Moon appears to subtend an angle of about 5.2°.

By convention, for magnifying glasses and optical microscopes, where the size of the object is a linear
dimension and the apparent size is an angle, the magnification is the ratio between the apparent (angular)
size as seen in the eyepiece and the angular size of the object when placed at the conventional closest
distance of distinct vision: 25 cm from the eye.

By instrument

Single lens
The linear magnification of a thin lens is

where is the focal length, is the distance from the lens to the
object, and as the distance of the object with respect
to the front focal point. A sign convention is used such that and
(the image distance from the lens) are positive for real object
and image, respectively, and negative for virtual object and
images, respectively. of a converging lens is positive while for a A Thin lens where black dimensions
diverging lens it is negative. are real, the greys are virtual.

For real images, is negative and the image is inverted. For

virtual images, is positive and the image is upright.

With being the distance from the lens to the image, the height of the image and the height of the
object, the magnification can also be written as:
Note again that a negative magnification implies an inverted image.

The image magnification along the optical axis direction , called longitudinal magnification, can also
be defined. The Newtonian lens equation is stated as , where and
as on-axis distances of an object and the image with respect to respective focal points, respectively.
is defined as

and by using the Newtonian lens equation,

The longitudinal magnification is always negative, means that, the object and the image move toward the
same direction along the optical axis. The longitudinal magnification varies much faster than the
transverse magnification, so the 3-dimensional image is distorted.

Photography
The image recorded by a photographic film or image sensor is always a real image and is usually
inverted. When measuring the height of an inverted image using the cartesian sign convention (where the
x-axis is the optical axis) the value for hi will be negative, and as a result M will also be negative.
However, the traditional sign convention used in photography is "real is positive, virtual is negative".[1]
Therefore, in photography: Object height and distance are always real and positive. When the focal
length is positive the image's height, distance and magnification are real and positive. Only if the focal
length is negative, the image's height, distance and magnification are virtual and negative. Therefore, the
photographic magnification formulae are traditionally presented as[2]

Magnifying glass
The maximum angular magnification (compared to the naked eye) of a magnifying glass depends on how
the glass and the object are held, relative to the eye. If the lens is held at a distance from the object such
that its front focal point is on the object being viewed, the relaxed eye (focused to infinity) can view the
image with angular magnification
Here, is the focal length of the lens in centimeters. The constant 25 cm is an estimate of the "near point"
distance of the eye—the closest distance at which the healthy naked eye can focus. In this case the
angular magnification is independent from the distance kept between the eye and the magnifying glass.

If instead the lens is held very close to the eye and the object is placed closer to the lens than its focal
point so that the observer focuses on the near point, a larger angular magnification can be obtained,
approaching

A different interpretation of the working of the latter case is that the magnifying glass changes the diopter
of the eye (making it myopic) so that the object can be placed closer to the eye resulting in a larger
angular magnification.

Microscope
The angular magnification of a microscope is given by

where is the magnification of the objective and the magnification of the eyepiece. The
magnification of the objective depends on its focal length and on the distance between objective
back focal plane and the focal plane of the eyepiece (called the tube length):

The magnification of the eyepiece depends upon its focal length and is calculated by the same
equation as that of a magnifying glass:

Note that both astronomical telescopes as well as simple microscopes produce an inverted image, thus the
equation for the magnification of a telescope or microscope is often given with a minus sign.

Telescope
The angular magnification of an optical telescope is given by
in which is the focal length of the objective lens in a refractor or of the primary mirror in a reflector,
and is the focal length of the eyepiece.

Measurement of telescope magnification

Measuring the actual angular magnification of a telescope is difficult, but it is possible to use the
reciprocal relationship between the linear magnification and the angular magnification, since the linear
magnification is constant for all objects.

The telescope is focused correctly for viewing objects at the distance for which the angular magnification
is to be determined and then the object glass is used as an object the image of which is known as the exit
pupil. The diameter of this may be measured using an instrument known as a Ramsden dynameter which
consists of a Ramsden eyepiece with micrometer hairs in the back focal plane. This is mounted in front of
the telescope eyepiece and used to evaluate the diameter of the exit pupil. This will be much smaller than
the object glass diameter, which gives the linear magnification (actually a reduction), the angular
magnification can be determined from

Maximum usable magnification

With any telescope, microscope or lens, a maximum magnification exists beyond which the image looks
bigger but shows no more detail. It occurs when the finest detail the instrument can resolve is magnified
to match the finest detail the eye can see. Magnification beyond this maximum is sometimes called
"empty magnification".

For a good quality telescope operating in good atmospheric conditions, the maximum usable
magnification is limited by diffraction. In practice it is considered to be 2× the aperture in millimetres or
50× the aperture in inches; so, a 60 mm diameter telescope has a maximum usable magnification of 120×.

With an optical microscope having a high numerical aperture and using oil immersion, the best possible
resolution is 200 nm corresponding to a magnification of around 1200×. Without oil immersion, the
maximum usable magnification is around 800×. For details, see limitations of optical microscopes.

Small, cheap telescopes and microscopes are sometimes supplied with the eyepieces that give
magnification far higher than is usable.

The maximum relative to the minimum magnification of an optical system is known as zoom ratio.

"Magnification" of displayed images

Magnification figures on pictures displayed in print or online can be misleading. Editors of journals and
magazines routinely resize images to fit the page, making any magnification number provided in the
figure legend incorrect. Images displayed on a computer screen change size based on the size of the
screen. A scale bar (or micron bar) is a bar of stated length superimposed on a picture. When the picture
is resized the bar will be resized in proportion. If a picture has a scale bar, the actual magnification can
easily be calculated. Where the scale (magnification) of an image is important or relevant, including a
scale bar is preferable to stating magnification.

See also
Lens
Magnifying glass
Microscope
Optical telescope
Screen magnifier

References
1. Ray, Sidney F. (2002). Applied Photographic Optics: Lenses and Optical Systems for
Photography, Film, Video, Electronic and Digital Imaging (https://books.google.com/books?i
d=cuzYl4hx-B8C&pg=PA40). Focal Press. p. 40. ISBN 0-240-51540-4.
2. Kingslake, Rudolph (1992). Optics in Photography. Bellingham, Washington: SPIE Optical
Engineering Press. p. 32. ISBN 0-8194-0763-1. "If a lens is thin, or if we can guess at the
position of the principal planes, we can readily construct from [1/di + 1/do = 1/f and
M = di/do] the following simple rules that it is well to bear in mind. They refer specifically to
the case of a positive lens forming a real image of a real object, all distances and the
magnification being assumed to be positive quantities. If virtual images are involved, it is
better to return to the original formulas, [previously stated]. The equations are
[do = f(1 + 1/M) and di = f(1 + M)]."

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