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Paraxial rays are found to focus far away the lens whereas marginal rays focus near to the lens, why? I didn't find satisfactory answer in my academic book..

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A lens with spherical surfaces (i.e. surfaces that are a segment of a sphere) does not focus all parallel light rays onto a single point. Instead there is a focal surface called a caustic. This causes spherical aberation.

As long as the lens curvature is small, i.e. the ray is near to the optical axis (paraxial rays), the aberation is small and we approximate the caustic by a single point. However far from the axis the aberation is significant and the rays come to a focus at a different point.

As far as I know there is no simply way to show this. You'd need to sit down and calculate the path for the light ray as a function of distance from the optical axis, which isn't that difficult but is tedious.

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The formula for the focus of marginal rays is given by $R - \frac{R}{\cos{\theta}}$ where $\theta$ is the angle made by the marginal ray with the line joining it's point of incidence with centre of curvature. We can see that for marginal rays $\theta$ is large hence the focus is small. Thus marginal rays converge close to the lens (for concave lenses) and diverge from a nearer point (for convex lenses). In fact, for paraxial rays $\theta$ is approximately $0$ and so $F = R -\frac{R}{\cos(0)} = R/2$

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I texified your equations however the last one is obviously wrong since $\cos(0) = 1$. –  Brandon Enright May 29 '13 at 6:27
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