These mirrors are called "converging mirrors" because they tend to collect light that falls on them, refocusing parallel incoming rays toward a focus. This is because the light is reflected at different angles at different spots on the mirror as the normal to the mirror surface differs at each spot.

Concave mirrors are used in reflecting telescopes.^[5] They are also used to provide a magnified image of the face for applying make-up or shaving.^[6] In illumination applications, concave mirrors are used to gather light from a small source and direct it outward in a beam as in torches, headlamps and spotlights, or to collect light from a large area and focus it into a small spot, as in concentrated solar power. Concave mirrors are used to form optical cavities, which are important in laser construction. Some dental mirrors use a concave surface to provide a magnified image. The mirror landing aid system of modern aircraft carriers also uses a concave mirror.

Concave mirror image

Analysis

Mirror equation, magnification, and focal length

The Gaussian mirror equation, also known as the mirror and lens equation, relates the object distance ${\displaystyle d_{\mathrm {o} }}$ and image distance ${\displaystyle d_{\mathrm {i} }}$ to the focal length ${\displaystyle f}$:^[2]

${\displaystyle {\frac {1}{d_{\mathrm {o} }}}+{\frac {1}{d_{\mathrm {i} }}}={\frac {1}{f}}}$.

The sign convention used here is that the focal length is positive for concave mirrors and negative for convex ones, and ${\displaystyle d_{\mathrm {o} }}$ and ${\displaystyle d_{\mathrm {i} }}$ are positive when the object and image are in front of the mirror, respectively. (They are positive when the object or image is real.)^[2]

For convex mirrors, if one moves the ${\displaystyle 1/d_{\mathrm {o} }}$ term to the right side of the equation to solve for ${\displaystyle 1/{}_{}}$

The Gaussian mirror equation, also known as the mirror and lens equation, relates the object distance ${\displaystyle d_{\mathrm {o} }}$ and image distance ${\displaystyle d_{\mathrm {i} }}$ to the focal length ${\displaystyle f}$:^[2]

${\displaystyle \frac{1}{d_{\mathrm{o}}}+}$

The sign convention used here is that the focal length is positive for concave mirrors and negative for convex ones, and ${\displaystyle d_{\mathrm {o} }}$ and ${\displaystyle d_{\mathrm {i} }}$ are positive when the object and image are in front of the mirror, respectively. (They are positive when the object or image is real.)^[2]

For convex mirrors, if one moves the ${\displaystyle 1/d_{\mathrm {o} }}$ term to the right side of the equation to solve for ${$

For convex mirrors, if one moves the ${\displaystyle 1/d_{\mathrm {o} }}$ term to the right side of the equation to solve for ${\displaystyle 1/d_{\mathrm {i} }}$, then the result is always a negative number, meaning that the image distance is negative—the image is virtual, located "behind" the mirror. This is consistent with the behavior described above.

For concave mirrors, whether the image is virtual or real depends on how large the object distance is compared to the focal length. If the ${\displaystyle 1/f}$ term is larger than the ${\displaystyle 1/d_{\mathrm {o} }}$ term, then ${\displaystyle 1/d_{\mathrm {i} }}$ is positive and the image is real. Otherwise, the term is negative and the image is virtual. Again, this validates the behavior described above.

The magnification of a mirror is defined as the height of the image divided by the height of the object:

By convention, if the resulting magnification is positive, the image is upright. If the magnification is negative, the image is inverted (upside down).

Ray tracing

Main article: Ray tracing (physics)

The image location and size can also be found by graphical ray tracing, as illustrated in the figures above. A ray drawn from the top of the object to the mirror surface vertex (where the optical axis meets the mirror) will form an angle with the optical axis. The reflected ray has the same angle to the axis, but on the opposite side (See Specular reflection).

A second ray can be drawn from the top of the object, parallel to the optical axis. This ray is reflected by the mirror and passes through its focal point. The point at which these two rays meet is the image point corresponding to the top of the object. Its distance from the optical axis defines the height of the image, and its location along the axis is the image location. The mirror equation and magnification equation can be derived geometrically by considering these two rays. A ray that goes from the top of the object through the focal point can be considered instead. Such a ray reflects parallel to the optical axis and also passes through the image point corresponding to the top of the object.

Ray transfer matrix of spherical mirrors

The image location and size can also be found by graphical ray tracing, as illustrated in the figures above. A ray drawn from the top of the object to the mirror surface vertex (where the optical axis meets the mirror) will form an angle with the optical axis. The reflected ray has the same angle to the axis, but on the opposite side (See Specular reflection).

A second ray can be drawn from the top of the object, parallel to the optical axis. This ray is reflected by the mirror and passes through its focal point. The point at which these two rays meet is the image point corresponding to the top of the object. Its distance from the optical axis defines the height of the image, and its location along the axis is the image location. The mirror equation and magnification e

A second ray can be drawn from the top of the object, parallel to the optical axis. This ray is reflected by the mirror and passes through its focal point. The point at which these two rays meet is the image point corresponding to the top of the object. Its distance from the optical axis defines the height of the image, and its location along the axis is the image location. The mirror equation and magnification equation can be derived geometrically by considering these two rays. A ray that goes from the top of the object through the focal point can be considered instead. Such a ray reflects parallel to the optical axis and also passes through the image point corresponding to the top of the object.

The mathematical treatment is done under the paraxial approximation, meaning that under the first approximation a spherical mirror is a parabolic reflector. The ray matrix of a concave spherical mirror is shown here. The ${\displaystyle C}$ element of the matrix is ${\displaystyle -{\frac {1}{f}}}$, where ${\displaystyle f}$ is the focal point of the optical device.

Boxes 1 and 3 feature summing the angles of a triangle and comparing to π radians (or 180°). Box 2 shows the

Boxes 1 and 3 feature summing the angles of a triangle and comparing to π radians (or 180°). Box 2 shows the Maclaurin series of ${\displaystyle \arccos \left(-{\frac {r}{R}}\right)}$ up to order 1. The derivations of the ray matrices of a convex spherical mirror and a thin lens are very similar.