Two-mirror foundation

A telescope with only one curved mirror, such as a Newtonian telescope, will always have aberrations. If the mirror is spherical, it will suffer primarily from spherical aberration. If the mirror is made parabolic, to correct the spherical aberration, then it still suffers from coma and astigmatism, since there are no additional design parameters one can vary to eliminate them. With two non-spherical mirrors, such as the Ritchey–Chrétien telescope, coma can be eliminated as well, by making the two mirrors' contribution to total coma cancel. This allows a larger useful field of view. However, such designs still suffer from astigmatism.

The basic Ritchey–Chrétien two-surface design is free of third-order coma and spherical aberration.^[1] However, the two-surface design does suffer from fifth-order coma, severe large-angle astigmatism, and comparatively severe field curvature.^[2]

Further corrections by a third element

When focused midway between the sagittal and tangential focusing planes, stars appear as circles, making the Ritchey–Chrétien well suited for wide field and photographic observations. The remaining aberrations of the two-element basic design may be improved with the addition of smaller optical elements near the focal plane.^[3][4]

Astigmatism can be cancelled by including a third curved optical element. When this element is a mirror, the result is a three-mirror anastigmat. Alternatively, a RCT may use one or several low-power lenses in front of the focal plane as a field-corrector to correct astigmatism and flatten the focal surface, as for example the SDSS telescope and the VISTA telescope; this can allow a field-of-view up to around 3° diameter.

The Schmidt camera can deliver even wider fields up to about 7°. However, the Schmidt requires a full-aperture corrector plate, which restricts it to apertures below 1.2 meters, while a Ritchey–Chrétien can be much larger. Other telescope designs with front-correcting elements are not limited by the practical problems of making a multiply-curved Schmidt corrector plate, such as the Lurie–Houghton design.

Aperture obstruction

In a Ritchey–Chrétien design, as in most Cassegrain systems, the secondary mirror blocks a central portion of the aperture. This ring-shaped entrance aperture significantly reduces a portion of the modulation transfer function (MTF) over a range of low spatial frequencies, compared to a full-aperture design such as a refractor.^[5] This MTF notch has the effect of lowering image contrast when imaging broad features. In addition, the support for the secondary (the spider) may introduce diffraction spikes in images.

Mirror

Diagram of a Ritchey–Chrétien reflector telescope

The radii of curvature of the primary and secondary mirrors, respectively, in a two-mirror Cassegrain configuration are:

${\displaystyle R_{1}=-{\frac {2DF}{F-B}}=-{\frac {2F}{M}}}$

and

${\displaystyle R_{2}=-{\frac {2DB}{F-B-D}}=-{\frac {2B}{M-1}}}$,

where

• ${\displaystyle F}$ is the effective focal length of the system,
• ${\displaystyle B}$ is the back focal length (the distance from the secondary to the focus),
• ${\displaystyle D}$ is the distance between the two mirrors and
• ${\displaystyle M=(F-B)/D}$ is the secondary magnification.^[6]

If, instead of ${\displaystyle B}$ and ${\displaystyle D}$, the known quantities are the focal length of the primary mirror, ${\displaystyle f_{1}}$, and the distance to the focus behind the primary mirror, ${\displaystyle b}$, then ${\displaystyle D=f_{1}(F-b)/(F+f_{1})}$ and ${\displaystyle B=D+b}$.

For a Ritchey–Chrétien system, the conic constants ${\displaystyle K_{1}}$ and ${\displaystyle K_{2}}$ of the two mirrors are chosen so as to eliminate third-order spherical aberration and coma; the solution is:

${\displaystyle K_{1}=-1-{\frac {2}{M^{3}}}\cdot {\frac {B}{D}}}$

and

${\displaystyle K_{2}=-1-{\frac {2}{(M-1)^{3}}}\left[M(2M-1)+{\frac {B}{D}}\right]}$.

Note that ${\displaystyle K_{1}}$ and ${\displaystyle K_{2}}$ are less than ${\displaystyle -1}$ (since ${\displaystyle M>1}$), so both mirrors are hyperbolic. (The primary mirror is typically quite close to being parabolic, however.)

The hyperbolic curvatures are difficult to test, especially with equipment typically available to amateur telescope makers or laboratory-scale fabricators; thus, older telescope layouts predominate in these applications. However, professional optics fabricators and large research groups test their mirrors with interferometers. A Ritchey–Chrétien then requires minimal additional equipment, typically a small optical device called a null corrector that makes the hyperbolic primary look spherical for the interferometric test. On the Hubble Space Telescope, this device was built incorrectly (a reflection from an un-intended surface leading to an incorrect measurement of lens position) leading to the error in the Hubble primary mirror.^[7]

Incorrect null correctors have led to other mirror fabrication errors as well, such as in the New Technology Telescope.

Extra flat mirrors

In practice, each of these designs may also include any number of flat fold mirrors, used to bend the optical path into more convenient configurations. This article only discusses the mirrors required for forming an image, not those for placing it in a convenient location.