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