Light · reflection
Mirrors
A mirror has one rule: the ray leaves at the same angle to the normal as it arrived. Curve the mirror and the normals tip, and parallel light gathers to a focus. A sphere gathers it almost to a point. A parabola gathers it exactly, but only for light along its axis.
The mirror
Measured
Spherical mirrors
For rays close to the axis, a spherical mirror of radius R focuses parallel light halfway between itself and its centre of curvature C:
A ray striking at height h, where the radius from C makes the angle α with the axis (sin α = h/R), crosses the axis a distance R/(2 cos α) from C. That is nearer the mirror than F, so a wide spherical mirror has spherical aberration like a lens. A convex mirror spreads light as if from a virtual focus behind it. That gives a wide view in a small mirror, at the price of images that look further away than they are.
The parabola
A parabola is the curve whose every point is as far from the focus as from a fixed line behind it. That makes every ray parallel to the axis reflect through the focus, however far from the axis it strikes. Choose the parabolic mirror: the spherical aberration reads zero. Then tilt the light a few degrees. The rays no longer meet at one point but cross unevenly, off to one side. Seen in a real telescope, where the cross-section becomes a whole image, that error is a comet-shaped blur called coma, and it limits the useful field of a parabolic telescope. Used the other way round, a lamp at the focus of a parabola sends out a parallel beam, as in a searchlight. A dish aerial gathers radio waves the same way.
Isaac Newton’s first reflecting telescope (1668) used a spherical mirror of polished speculum metal. John Hadley showed the Royal Society a reflector with a true parabolic mirror in 1721, and parabolic main mirrors served most large telescopes until the mid-20th century, the 5 m Hale telescope (1948) among them. Most large telescopes since, from Hubble to Keck, use a pair of hyperbolic mirrors (the Ritchey–Chrétien design) that cancels coma as well. A mirror cannot have chromatic aberration, because reflection treats every colour the same. That is one reason the largest telescopes are reflectors.
These are rays, the geometrical-optics picture: light is treated as lines that bend and split at surfaces. Diffraction and interference, which need the wave picture, are not shown. The bench is a flat cross-section, so a lens here is really a cylinder of glass seen end-on. That is exact for rays in the plane of the drawing. Air is taken as n = 1, and the glass and water indices are given relative to air, so the angles are right. Colours on screen are an approximation of the spectrum.