1.3. Optical system of a telescope

FIGURE 5A: Image formed by a telescope mirror objective, as seen without the eyepiece (flat is omitted for simplicity). Wavefronts emitted from distant object of height h become practically flat over the aperture of a telescope, here a concave mirror M. The mirror changes the shape of the incoming flat wavefront section (W) into spherical (W') by delaying reflection of the points in phase belonging to the wavefront's inner area. Point of convergence (c) - or focal point - is at the center of curvature R of the wavefront. Thus, with the stop at mirror surface and exit pupil plane at mirror vertex, the mirror focal length equals the radius of the wavefront in the pupil.  The top point of distant object h, at an angle α from the optical axis, is imaged into (reversed) top of the image h' by off-axis wavefront Wa, originating at the object's top point. If observed directly, from the least distance of distinct vision v (approx. 25cm, or 10 inches) most of the light from the object image h' misses the eye pupil (more so for the points farther off-axis, with no light from the image's reversed top reaching the eye); also, magnification is limited to ƒ/v, ƒ being the objective's focal length. The apparent image angle β determines objective magnification as Mo=tanβ/tanα.

FIGURE 5B: Image formed by a telescope lens objective, as seen through the eyepiece. Since light is slowed down in glass, the in-phase points of the incident axial wavefront W are retarded the most in the center of the lens objective L (for simplicity, both objective and eyepiece are shown as a single lens), and the least at its edges. Properly made lens objective will re-shape flat incident wavefront W into spherical (W') after exiting the lens. That is a goal for off-axis point wavefronts (Wa) as well, although some form of deviation due to tilt-created asymmetry is usually present. In terms of rays, change in direction of straight lines orthogonal to the wavefront (rays), resulting from its new shape, is called refraction. The lens objective focal length ƒO is the distance between its second principal point P2 and the focal point F. The eyepiece (EP), placed at a distance of its focal length ƒE from the object image h' formed by the objective, converts diverging spherical wavefront into flat, for which the eye has preference. It also increases the apparent incidence angle (α vs. ε), making the object imaged at the retina (E) appear larger by a factor ~ƒO/ƒE. Use of an eyepiece allows for far more light from the object's image (all of it, if properly designed) to reach the eye, much higher magnifications, and much wider fields compared to observing image formed by the objective with eye alone.

FIGURE 6: Geometry of image formation by thin lens (A) and mirror (B) in air. Incident ray parallel to the optical axis (2) is directed, after reflection or refraction, to the focal point F, located at its intersection with the optical axis. It determines the focal length ƒ. Incident ray coming from the same object point through the front focus F' or F (3) refracts or reflects parallel to the optical axis; its intersection with ray 2 determines the image point location. Alternatively, it is also determined by the point of intersection with the incident ray arriving at the center of the objective (1, the chief ray). As object distance increases, incident rays coming through the front focus and center of the objective (3 and 1, respectively) merge closer, practically merging together for very distant objects. At that point, image magnification, given as image-to-object-distance ratio, approaches zero - with the field angle α reduced to a very small, but finite quantity - and the image practically forms at the focal point location. When lens thickness is significant with respect to the object distance and focal length, the ray path through the lens becomes a factor in determining lens' focal length ƒ, and needs to be taken into account (C). Here, focal length equals the separation between 2nd principal plane - a plane normal to the axis, containing the point of intersection (principal point P2) of extended path of a collimated incident ray and reversed path of it after exiting the lens - and the focal point (F'). It is preceded by the 1st principal plane, determined in the same manner with collimated incident ray from the opposite direction (principal point P1). The corresponding points on the two principal planes are always at the same separation from axis, i.e. lay on a line parallel to it; in effect, all rays refracted by a lens behave as if the only refraction is taking place at the principal plane. A ray whose incident and final orientation doesn't change (in other words, its path before and after lens are parallel) determines lens' nodal points. For a single lens, nodal points lay in the principal planes, 1st nodal point (N) in the 1st principal plane, and 2nd nodal point (N') in the 2nd principal plane. Principal planes are not necessarily contained within lens, and may be located at a significant distance from it (for instance, with Maksutov corrector). Also, in unequifocal lenses or systems, such as human eye, with different incident and final medium refractive index (thus different focal lengths in these respective media), nodal points are displaced axially from the principal planes, although the nodal points separation remains identical to that of the principal planes, Note that the above scheme is a paraxial (Gaussian) idealization, ignoring lens' aberrations, thus only valid for paraxial rays.