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I saw this posted on the forum with an answer: Cause of Spherical Aberrations

However, the answer helps explain aberration in lenses. Why is there spherical aberration in concave mirrors?

Thanks, -Prasad

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Prasad
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    Please find some basic books or web pages on lenses. A mirror is exactly the same as a simple plan0-convex lens in first approximation. – Carl Witthoft Apr 13 '16 at 19:37
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    I'm voting to close this question as off-topic because it shows no research effort. – ACuriousMind Apr 14 '16 at 00:38
  • I know it's been a long time but if you're looking for a proof of the existence of spherical aberration in concave lenses, maybe this post of mine would help you : https://physics.stackexchange.com/questions/563003/proof-that-all-rays-parallel-to-the-principal-axis-meet-at-the-focus-during-refl#563006 It was an attempt to prove that all rays in a concave mirror meet at the focus, but I ended up re-discovering that spherical aberration exists. – Rajdeep Sindhu Jul 17 '20 at 06:21

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It just geometry. If you want all incident rays parallel to the principal axis to reflect through a single point the mirror needs to parabolic in shape. For a focal length f the equation of the parabola (opening upwards) would be $4fy=x^{2}$.

A spherical mirror will have approximately the same shape as a parabolic mirror near the vertex. The focal point will be at a distance of half the radius of curvature from the vertex. However, with a spherical mirror, as incident rays get farther from the principal axis they produce reflected rays that cross the principal axis at points nearer to the mirror than half the radius. This is the origin of the spherical aberration.

Spherical aberration in concave mirror

Image source:
Aberration de sphéricité d'un miroir sphérique concave by Jean-Jacques MILAN, GNU Free Documentation License.

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