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Suppose a ray of light hits a concave mirror and is parallel to the principal axis but far away from it such that it doesn't follow paraxial ray approximation. Will it pass through focus or between focus and radius of curvature or between pole and focus?

Here pole, focus and radius of curvature mean the same thing as in paraxial ray approximation .

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It is depending upon the distance of ray from axis of mirror. – ABC Apr 27 '13 at 16:28
Removed the ridiculous down-vote by adding a +1. – Antillar Maximus May 15 '13 at 11:31
It really depends on the shape of your "concave" mirror. Is it spherical, parabolic, or just any old dent in a shiny surface? It affects the answer... – Floris Oct 17 '14 at 19:40

3 Answers 3

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enter image description here

Assuming mirror to be spherical section. C is the center of sphere.

See, Using trigonometry. $$x=d \times \sin(2\theta)$$ $$x=R\times\sin\theta$$

Eliminate $\theta$ and get $d$ : distance from Center of curvature as a function of $x$.

Verify for small theta where $\sin\theta\approx\theta$

If you just want to see that which side ray bents then see. $$d=\dfrac{R\times \sec\theta}2$$ which shows that $d\ge R/2$ . So, ray bends towards the pole as looses it paraxial character.

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Sorry for the rough image made in Paint . :P – ABC Apr 27 '13 at 16:40

A parabolic mirror is a special case of a concave mirror. The rays at the rim are refracted to the focal point. This is true in simplification of geometric optics and perfect manufactured mirror. However in modern techiques like injection molding there are more imperfections at the rims of mirrors.

The question of a general concave mirror can be answered depending on the deviation of curvature $\frac{1}{R}$. If deviation is positive, then the rays are closer on optical axis.

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For a parabolic mirror, the answer is "all rays go through the focus". For a spherical mirror, you can see the answer most easily by looking at the limit as the spherical mirror is almost a hemisphere:

enter image description here

Clearly the rays that start further off axis are focused closer to the mirror.

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