Magnification

right|thumb|The postage stamp appears larger with the use of a [[magnifying glass.]]

thumb|thumbtime=0|Stepwise magnification by 6% per frame into a 39-megapixel image. In the final frame, at about 170x, an image of a bystander is seen reflected in the man's [[cornea.]]

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 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.
• A telescope, which uses its large objective lens or primary mirror to create an image of a distant object and then allows the user to examine the image closely with a smaller eyepiece lens, thus making the object look 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 an optical instrument with an eyepiece as an example, the linear dimension of an image seen through the eyepiece cannot be given if it is an virtual image at an infinite distance, thus size in this case may mean the angle subtended between an edge (or both edges, depending on the definition) of the image and the optical axis of the instrument (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

<math display="block">M_A=\frac{\tan \varepsilon}{\tan \varepsilon_0}\approx \frac{\varepsilon}{ \varepsilon_0}</math>

where <math display="inline">\varepsilon_0</math> is the angle subtended by an object (w.r.t the optical axis) and <math display="inline">\varepsilon</math> is the angle subtended by its image (also w.r.t the optical axis) made by an optical instrument.

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 an object is a linear dimension and the apparent size (the image size) of it is an angle, the magnification is the ratio between the apparent (angular) size as seen via instrument and the angular size of the object when the object is placed at the conventional closest distance of distinct vision to an unaided human eye: from the eye (called the near point).

thumb|A [[thin lens where black dimensions are real, the greys are virtual.]]

By instrument

Single lens

The linear magnification of a thin lens is

<math display="block">M = {f \over f-d_\mathrm{o = - \frac{f}{x_o}</math>

where <math display="inline">f</math> is the focal length, <math display="inline">d_\mathrm{o}</math> is the distance from the lens to the object, and <math display="inline">x_0 = d_0 - f</math> as the distance of the object with respect to the front focal point. A sign convention is used such that <math display="inline">d_0</math> and <math>d_i</math> (the image distance from the lens) are positive for real object and image, respectively, and negative for virtual object and images, respectively. <math display="inline">f</math> of a converging lens is positive while for a diverging lens it is negative.

For real images, <math display="inline">M</math> is negative and the image is inverted. For virtual images, <math display="inline">M</math> is positive and the image is upright.

With <math display="inline">d_\mathrm{i}</math> being the distance from the lens to the image, <math display="inline">h_\mathrm{i}</math> the height of the image and <math display="inline">h_\mathrm{o}</math> the height of the object, the magnification can also be written as Therefore, in photography: Object height and distance are always and positive. When the focal length is positive the image's height, distance and magnification are and positive. Only if the focal length is negative, the image's height, distance and magnification are and negative. Therefore, the ' formulae are traditionally presented as

<math display="block">\begin{align}

M &= {d_\mathrm{i} \over d_\mathrm{o = {h_\mathrm{i} \over h_\mathrm{o \\

&= {f \over d_\mathrm{o}-f} = {d_\mathrm{i}-f \over f}

\end{align}</math>

Magnifying glass

The angular magnification of a magnifying glass is defined as the ratio of an angle ε that the image (made by the glass) of an object located at the near point (typically 25 cm away from a human eye) subtends on the retina of the eye to an angle ε₀ that the object (at the same location) subtends on the retina without the glass., and (2) a circular shape of the retina; the twice angle between the eye's optical axis and the edge of the image on the retina (circularly shaped) is translated to twice larger the recognized image.

For a magnified and erected image of an object, the object needs to be located within the focal length of a converging lens (see the table Images of Real Objects Formed by Thin Lenses). The angular magnification of a converging lens as a magnifying glass, depends on how the glass and the object are located, relative to the eye.

The angular magnification M_A = ε/ε₀ can be, in paraxial approximation where tan(ε) ≈ ε, expressed as (h_i/L_i)/(h_o/L_N) = (h_iL_N)/(h_oL_i) where h_o is for the object height (w.r.t the optical axis), h_i for the image height (also w.r.t the axis), L_N for the near point distance from the eye (along the optical axis), and L_i is the image distance from the eye (also along the axis).

By using the transverse magnification M = h_i/h_o = -d_i/d_o, M_A = -(d_iL_N)/(d_oL_i). By using the thin lens equation 1/d_o + 1/d_i = 1/f (f as the focal length of the lens), M_A = (1 - d_i/f)(L_N/L_i). Because L_i = L_l - d_i (d_i is negative for a virtual image, made by a converging lens as a magnifying glass) so d_i = L_l - L_i, M_A becomes

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