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Photometry and Geometrical Optics

Problem 5.30

A parallel beam of light falls from vacuum on a surface enclosing a medium with refractive index \(n\) (Fig. 5.6 ). Find the shape of that surface, \(x(r),\) if the beam is brought into focus at the point \(F\) at a distance \(f\) from the crest \(O\). What is the maximum radius of a beam that can still be focussed?

Reveal Answer
x=nfn+1(1−1−(n+1)r2(n−1)f2),rmax=f(n−1)/(n+1)x=\frac{n f}{n+1}\left(1-\sqrt{1-\frac{(n+1) r^{2}}{(n-1) f^{2}}}\right), \quad r_{m a x}=f \sqrt{(n-1) /(n+1)}
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