why are mirrors used in search lights parabolic and not concave spherical?
A parabolic concave mirror is the ideal shape for focusing light to a point. This is useful in taking parallel light waves and focusing them to a point (like in a telescope), or taking light from a point source and making the waves parallel (like a searchlight).
Optical mirrors are typically made by orbital grinding, where a grinding surface is moved in circles over a flat mirror to grind it into a curved mirror. The problem is that this produces a mirror that has a spherical curve instead of a parabolic curve. They look very much alike, but a spherical mirror does not quite focus light to a point--the focus is spread out a bit, resulting in imperfect focus. For most applications, spherical mirrors are pretty close, but in telescopes they will produce some distortion in the final image, and in a searchlight it will cause the beam to lose focus after a certain distance.
A true parabolic mirror is hideously difficult and expensive to fabricate compared to spherical mirrors. Making the parabolic main mirror in the Hubble Space Telescope required reexamination of the mirror between successive grindings to get the right shape, and cost millions of dollars. Obviously, for most amateur telescopes, spherical mirrors are sufficient and more cost efficient.
Likewise, for searchlights that are meant for short distances, a spherical mirror may be sufficient, but for long distances a parabolic mirror is required.
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