Magnification - AbsoluteAstronomy.com

Magnification is the process of enlarging something only in appearance, not in physical size. This enlargement is quantified by a calculated number also called "magnification". When this number is less than one it refers to a reduction in size, sometimes called "minification" or "de-magnification".

Typically magnification is related to scaling up visuals or image

Image

An image is an artifact, for example a two-dimensional picture, that has a similar appearance to some subject—usually a physical object or a person.-Characteristics:...

s to be able to see more detail, increasing resolution

Angular resolution

Angular resolution, or spatial resolution, describes the ability of any image-forming device such as an optical or radio telescope, a microscope, a camera, or an eye, to distinguish small details of an object...

, using optics

Optics

Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

, printing

Printing

Printing is a process for reproducing text and image, typically with ink on paper using a printing press. It is often carried out as a large-scale industrial process, and is an essential part of publishing and transaction printing....

techniques, or digital processing. In all cases, the magnification of the image does not change the perspective

Perspective (visual)

Perspective, in context of vision and visual perception, is the way in which objects appear to the eye based on their spatial attributes; or their dimensions and the position of the eye relative to the objects...

of the image.

Examples of magnification

A magnifying glass
Magnifying glass
A magnifying glass is a convex lens that is used to produce a magnified image of an object. The lens is usually mounted in a frame with a handle ....

which uses a positive (convex) lens
Lens (optics)
A lens is an optical device with perfect or approximate axial symmetry which transmits and refracts light, converging or diverging the beam. A simple lens consists of a single optical element...

to make things look bigger by allowing the user to hold them closer to his eye.
A telescope
Telescope
A telescope is an instrument that aids in the observation of remote objects by collecting electromagnetic radiation . The first known practical telescopes were invented in the Netherlands at the beginning of the 1600s , using glass lenses...

which uses its large objective lens
Objective (optics)
In an optical instrument, the objective is the optical element that gathers light from the object being observed and focuses the light rays to produce a real image. Objectives can be single lenses or mirrors, or combinations of several optical elements. They are used in microscopes, telescopes,...

to create an image of a distant object and then allows the user to examine the image closely with a smaller eyepiece
Eyepiece
An eyepiece, or ocular lens, is a type of lens that is attached to a variety of optical devices such as telescopes and microscopes. It is so named because it is usually the lens that is closest to the eye when someone looks through the device. The objective lens or mirror collects light and brings...

lens thus making the object look larger.
A microscope
Microscope
A microscope is an instrument used to see objects that are too small for the naked eye. The science of investigating small objects using such an instrument is called microscopy...

which makes a small object appear as a much larger object 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
Slide projector
A slide projector is an opto-mechanical device to view photographic slides. Slide projectors were common in the 1950s to the 1970s as a form of entertainment; family members and friends would gather to view slide shows...

which projects a large image of a small slide on a screen.

Magnification as a number (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.

Linear or transverse magnification — For real image
Real image
In optics, a real image is a representation of an object in which the perceived location is actually a point of convergence of the rays of light that make up the image. If a screen is placed in the plane of a real image the image will generally become visible on the screen...

s, such as images projected on a screen, size means a linear dimension (measured, for example, in millimeters or inch
Inch
An inch is the name of a unit of length in a number of different systems, including Imperial units, and United States customary units. There are 36 inches in a yard and 12 inches in a foot...

es).

Angular magnification — For optical instrument
Optical instrument
An optical instrument either processes light waves to enhance an image for viewing, or analyzes light waves to determine one of a number of characteristic properties.-Image enhancement:...

s with an eyepiece
Eyepiece
An eyepiece, or ocular lens, is a type of lens that is attached to a variety of optical devices such as telescopes and microscopes. It is so named because it is usually the lens that is closest to the eye when someone looks through the device. The objective lens or mirror collects light and brings...

, the linear dimension of the image seen in the eyepiece (virtual image
Virtual image
In optics, a virtual image is an image in which the outgoing rays from a point on the object always diverge. It will appear to converge in or behind the optical device . A simple example is a flat mirror where the image of oneself is perceived at twice the distance from oneself to the mirror...

in 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
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

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.

Example: The angular size of the full moon is 0.5°, in binoculars
Binoculars
Binoculars, field glasses or binocular telescopes are a pair of identical or mirror-symmetrical telescopes mounted side-by-side and aligned to point accurately in the same direction, allowing the viewer to use both eyes when viewing distant objects...

with 10x magnification it appears to subtend an angle of 5°, which is roughly 1/10 of the field of view of typical eyepiece
Eyepiece
An eyepiece, or ocular lens, is a type of lens that is attached to a variety of optical devices such as telescopes and microscopes. It is so named because it is usually the lens that is closest to the eye when someone looks through the device. The objective lens or mirror collects light and brings...

s.

By convention, for magnifying glass

Magnifying glass

A magnifying glass is a convex lens that is used to produce a magnified image of an object. The lens is usually mounted in a frame with a handle ....

es and optical microscope

Microscope

A microscope is an instrument used to see objects that are too small for the naked eye. The science of investigating small objects using such an instrument is called microscopy...

s, 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.

Optical magnification is sometimes referred to as "power" (for example "10× power"), although this can lead to confusion with optical power

Optical power

Optical power is the degree to which a lens, mirror, or other optical system converges or diverges light. It is equal to the reciprocal of the focal length of the device. The dioptre is the most common unit of measurement of optical power...

Calculating the magnification of optical systems

Single lens
Lens (optics)
A lens is an optical device with perfect or approximate axial symmetry which transmits and refracts light, converging or diverging the beam. A simple lens consists of a single optical element...

: The linear magnification of a thin lens
Thin lens
[Image:Lens1.svg|thumb|A lens can be considered a thin lens if d [Image:Lens1.svg|thumb|A lens can be considered a thin lens if d [Image:Lens1.svg|thumb|A lens can be considered a thin lens if d...

is

where

is the focal length

Focal length

The focal length of an optical system is a measure of how strongly the system converges or diverges light. For an optical system in air, it is the distance over which initially collimated rays are brought to a focus...

and

is the distance from the lens to the object. Note that for real image

Real image

In optics, a real image is a representation of an object in which the perceived location is actually a point of convergence of the rays of light that make up the image. If a screen is placed in the plane of a real image the image will generally become visible on the screen...

is negative and the image is inverted. For virtual image

Virtual image

In optics, a virtual image is an image in which the outgoing rays from a point on the object always diverge. It will appear to converge in or behind the optical device . A simple example is a flat mirror where the image of oneself is perceived at twice the distance from oneself to the mirror...

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.

Photography
Photography
Photography is the art, science and practice of creating durable images by recording light or other electromagnetic radiation, either electronically by means of an image sensor or chemically by means of a light-sensitive material such as photographic film...

: The image recorded by a photographic film
Photographic film
Photographic film is a sheet of plastic coated with an emulsion containing light-sensitive silver halide salts with variable crystal sizes that determine the sensitivity, contrast and resolution of the film...

or image sensor
Image sensor
An image sensor is a device that converts an optical image into an electronic signal. It is used mostly in digital cameras and other imaging devices...

is always a real image
Real image
In optics, a real image is a representation of an object in which the perceived location is actually a point of convergence of the rays of light that make up the image. If a screen is placed in the plane of a real image the image will generally become visible on the screen...

and is usually inverted. When measuring the height of an inverted image using the cartesian
Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

sign convention (where the x-axis is the optical axis) the value for h_i will be negative, and as a result M will also be negative. However, the traditional sign convention used in photography is "real
Real image
In optics, a real image is a representation of an object in which the perceived location is actually a point of convergence of the rays of light that make up the image. If a screen is placed in the plane of a real image the image will generally become visible on the screen...

is positive, virtual
Virtual image
In optics, a virtual image is an image in which the outgoing rays from a point on the object always diverge. It will appear to converge in or behind the optical device . A simple example is a flat mirror where the image of oneself is perceived at twice the distance from oneself to the mirror...

is negative". 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:

Telescope
Telescope
A telescope is an instrument that aids in the observation of remote objects by collecting electromagnetic radiation . The first known practical telescopes were invented in the Netherlands at the beginning of the 1600s , using glass lenses...

: The angular magnification is given by

where

is the focal length

Focal length

of the objective lens and

is the focal length of the eyepiece

Eyepiece

An eyepiece, or ocular lens, is a type of lens that is attached to a variety of optical devices such as telescopes and microscopes. It is so named because it is usually the lens that is closest to the eye when someone looks through the device. The objective lens or mirror collects light and brings...

Magnifying glass
Magnifying glass
A magnifying glass is a convex lens that is used to produce a magnified image of an object. The lens is usually mounted in a frame with a handle ....

: 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 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

Focal length

of the lens

Lens (optics)

A lens is an optical device with perfect or approximate axial symmetry which transmits and refracts light, converging or diverging the beam. A simple lens consists of a single optical element...

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 maginification 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
Microscope
A microscope is an instrument used to see objects that are too small for the naked eye. The science of investigating small objects using such an instrument is called microscopy...

: The angular magnification 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

Focal length

and on the distance

between objective back focal plane and the focal plane of the eyepiece

Eyepiece

(called the tube length):

The magnification of the eyepiece depends upon its focal length

and calculated by the same equation as that of a magnifying glass (above).

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

Plus and minus signs

The plus and minus signs are mathematical symbols used to represent the notions of positive and negative as well as the operations of addition and subtraction. Their use has been extended to many other meanings, more or less analogous...

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 Ramsden disc. The diameter of this may be measured using an instrument known as a Ramsden dynameter

Dynameter

A dynameter is an instrument that measures the magnification of a telescope. It is usually a double-image micrometer used to measure the diameter of the image of the object glass. The magnifying power is found by comparing the actual diameter of the image on the glass with the measured image...

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 Ramsden disc. 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 or microscope, there is a maximum magnification 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 widely considered to be 2× the aperture in millimetres or 50× the aperture in inches, so a 60mm diameter telescope has a maximum usable magnification of 120×.

With an optical microscope having a high numerical aperture

Numerical aperture

In optics, the numerical aperture of an optical system is a dimensionless number that characterizes the range of angles over which the system can accept or emit light. By incorporating index of refraction in its definition, NA has the property that it is constant for a beam as it goes from one...

and using oil immersion

Oil immersion

In light microscopy, oil immersion is a technique used to increase the resolution of a microscope. This is achieved by immersing both the objective lens and the specimen in a transparent oil of high refractive index, thereby increasing the numerical aperture of the objective lens.Immersion oils are...

, 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 eyepieces that give magnification far higher than is usable.

Examples of magnification

Magnification as a number (optical magnification)

Calculating the magnification of optical systems

Measurement of telescope magnification

Maximum usable magnification

See also