Generally, this image will be smaller than the original extended objects appearance to the eye and inverted. Under the assumption that objects are located very far away from the lens, we can use the approximation that the rays from all light sources are approximately parallel, hence the image distance from the lens, termed the focal length, is approximately the same for all objects.
We should note here that, for mirrors, only those whose reflective surface follows the curve of a parabola actually bring parallel rays of light to a common focal point (this is readily derivable from the definition of a parabolic curve). Spherical mirrors reflect non-paraxial rays, i.e. rays from the axis of the spherical surface, to a circular blur of points termed the circle of confusion. This effect is called spherical aberration. It is just one of several aberrations that have to be corrected for telescopes capable of optical measurement. The diagram below shows the effect for a spherical vs. a parabolic reflective surface.
We shall return to the topic of aberrations and how to correct for them shortly. For the moment, we continue with the general description of lenses and mirrors.
For either a lens or a mirror, an important parameter is the ratio of focal length to diameter. This is called the f ratio. If we term the focal length of the lens or mirror, f, and its diameter, d, then the f ratio is just f/d. This ratio quantifies the brightness of the image produced by the lens or mirror and the size of the image produced. A small f ratio, such as f/3, for example, gives a brighter image than a large f ratio like f/15. In other words, an f/15 lens or mirror spreads the image out over a larger angular region than a f/3 lens or mirror does, hence the image brightness decreases.
The linear size of the image of an extended object is also related to the f ratio. For example, the Moon subtends a 1/2 degree arc on the sky. If its image covers 10 cm at the focal surface of the lens or mirror, then the scale is 0.05 degrees/cm or 200 cm per degree. For the size, s, of an image at the focus corresponding to 1 degree in the sky, we find that s = 0.01745 f. In these units, s is in units of f per degree. It is usually referred to as the plate scale.
The essential characteristic of any telescope is its light-gathering capacity. This capacity is directly proportional to the square of the diameter of the objective (surface area for light-gathering is Pi/4 * diameter^2). Hence, light-gathering power for comparative purposes is just the ratio of diameters squared for two telescopes.
Of course, another important characteristic of a telescope is its resolving power or ability to clearly separate images from objects which are close together in terms of angular separation on the sky. Typically, the resolving power is expressed in terms of the angular separation that must exist between two points in order for them to be easily separated. For example, the repaired Hubble telescope has a resolving power of about 0.05 arc seconds (an arc second is 1/3600 of a degree). If diffraction is the only limitation on the resolving power (it usually isn't), then the resolving power depends on the wavelength of light being imaged and the diameter of the telescope objective. For a circular objective, the circular diffraction pattern produced by the objective for a point source of light gives its first minimum as
sin(theta) = 1.22(lambda)/d
with d being the diameter of the lens, lambda the wavelength
of light, and theta the angular spread of the diffraction
pattern from primary maximum to first minimum. Since astronomers
usually work in arc seconds rather than radians and the angles
for typical images are small enough so that sin(theta) ~ theta,
the diffraction limit on the resolving power is given as
theta = 1.22 * (206,265 arc-sec/rad) * lambda/d
= 252,000 * lambda/d
As an example, if we use the Hubble telescope objective diameter
of 2.3 meters and a typical wavelength for light of 550 nanometers,
we would find
theta = (2.52 x 10^{5}) * (5.5 x 10^{-7} m)/2.3 m = 0.06 arc-sec
This means that if the Hubble is pointed at two stars that are more
than 0.06 arc seconds apart, you will see two separate stellar images.Of course, the theoretical limit of the resolving power of a telescope is never reached on earth. The atmosphere limits the typical resolving power or resolution to something significantly larger than its diffraction-limit. Hence, the call for placing telescopes beyond earth's atmosphere. You can read more about this fascinating subject and the even more intriguing technology that is being used to circumvent it in the May and June, 1994 Sky and Telescope magazine articles on adaptive optics.
The magnifying power of a telescope is its third important characteristic. The magnification is determined by the ratio of the focal length of the objective to the focal length of the eyepiece. Since the eyepiece magnifies the image produced at the focus by the objective, you can double the magnification of a telescope by simply changing to an eyepiece with half the focal length. This is of no use once you are at the resolving limit of the telescope, however. Magnifying a fuzzy image only makes the fuzziness worse.
Isaac Newton thought it impossible to correct chromatic aberration and hence was the first to produce a telescope with a mirror for the objective instead of a lens. Since the angle of reflection does not depend on transmission through the glass, reflective telescopes have no chromatic aberration. Also, since light does not travel through the glass, the mirror can be supported by its entire back surface. One can also reduce the weight by making the back surface a honeycomb shape which reduces material and thereby weight while still leaving enough structural strength to ensure that the mirror maintains its shape. Depending on reflection, however, means that you have to have some means of viewing the reflected light. A number of solutions have been devised, all of which depend on a secondary mirror to reflect the image at the focus of the objective onto a refractive eyepiece. Below are several designs along with their names (each is named after its inventor).
The Newtonian telescope typically has an f ratio between f/4 and f/8. Cassegrain telescopes are the most compact reflecting telescopes with f ratios between f/7 and f/12. The Gregorian telescope is longer and less aberration-free than the Cassegrain but produces an erect image.
In the early years of this century, Bernhard Schmidt produced a catadioptric design which incorporates both reflecting and refracting elements in the objective. The advantage here is that the mirror is a spherical concave shape which has a wide field of view and no aberrations other than spherical. Spherical reflecting surfaces lead to spherical aberrations as shown previously, so to correct for this Schmidt introduced a correction plate. The correction plate sits in a plane that passes through the center of curvature and causes additional convergence near its center and divergence further out (see the figure below).
It compensates for the spherical aberration of the mirror without introducing appreciable chromatic aberration. With this design, telescopes can be constructed with f ratios as large as f/0.6!
As an interesting aside, we should note that nature long ago anticipated the advantages of catadioptric designs for use in eye systems. For example, the scallop has 60 eyes, each of which is equipped with a lens possessing a spherical mirror in back of the lens. The cornea converges light falling near its center and diverges it near its periphery. After reflection, the light is focussed onto light-sensitive cells in the front surface of the fovea. Since the eye is so small, the fovea lies too close to the lens for the light to be focussed onto it by the cornea and lens alone.