# magnification

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## magnification

## Magnification

A measure of the effectiveness of an optical system in enlarging or reducing an image. For an optical system that forms a real image, such a measure is the lateral magnification *m*, which is the ratio of the size of the image to the size of the object. If the magnification is greater than unity, it is an enlargement; if less than unity, it is a reduction.

The angular magnification is the ratio of the angles formed by the image and the object at the eye. In telescopes the angular magnification (or, better, the ratio of the tangents of the angles under which the object is seen with and without the lens, respectively) can be taken as a measure of the effectiveness of the instrument.

Magnifying power is the measure of the effectiveness of an optical system used in connection with the eye. The magnifying power of a spectacle lens is the ratio of the tangents of the angles under which the object is seen with and without the lens, respectively. The magnifying power of a magnifier or an ocular is the ratio of the size under which an object would appear when seen through the instrument at a distance of 10 in. or 250 mm (the distance of distinct vision) divided by the object size. *See* Lens (optics), Optical image

## Magnification

the ratio of a linear or angular dimension of an image produced by an optical system to the corresponding dimension of the object.

The most commonly used optical systems have an optical axis and are said to be axisymmetric. In characterizing axisymmetric systems, the following types of magnification are distinguished: lateral, angular, and longitudinal.

The lateral magnification β is the ratio of the length *l*’ of the image of a line segment perpendicular to the optical axis and the length *l* of the segment: β = *l’ll*. When β > 0, *l* and *l*’ have the same direction, and the image is said to be erect. When β < 0, *l* and *l*’ have opposite directions, and the image is said to be inverted. The image is reduced if ǀβǀ < 1 and enlarged if ǀβǀ > 1. The value of β is used to characterize, for example, cameras.

The angular magnification γ is the ratio of the tangent of the angle of inclination *u*’ of a ray to the optical axis in the image space and the tangent of the angle of inclination *u* of the conjugate ray in the object space: γ = tan *u*’/tan *u*. The value of γ is an important characteristic of many optical instruments, such as magnifiers and eyepieces.

The longitudinal magnification α is the ratio of the length Δ*x*’ of a line segment lying along the optical axis in the image space and the length *Ax* of the conjugate segment in the object space: a = Δ*x*’/Δ*x*.

The relation between the quantities α, β, and γ is given by the equation αγ = β. If *n* and *n*’ are the indexes of refraction of the media in the object space and in the image space, respectively, then βγ = *n/n*’. For an optical system in air, *n* = *n*’ and γ = 1/β—that is, the angular magnification is inversely proportional to the lateral magnification. Consequently, the greater the lateral magnification, the narrower the light beams with which the image is formed and the lower its illuminance. The relation between α and β is given by the equation α (*n*’/*n*)β^{2}; when *n* = *n*’, α = β^{2}.

### REFERENCES

Landsberg, G. S.*Optika*, 4th ed. (

*Obshchii kurs fiziki*, vol. 3.) Moscow, 1957.

Tudorovskii, A. I.

*Teoriia opticheskikh priborov*, vol. 1, 2nd ed. Moscow-Leningrad, 1948.