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Physics Lenses System and Laser Problem?

Two positive lenses are to be used as a laser beam expander. An axial 1.0 mm diameter beam enters the short focal length positive lens, which is followed by a somewhat longer focal length positive lens, from which it emerges with a diameter of 8.0 mm. Given that the first lens has a 50.0 mm focal length, determine the focal length of the second lens and the separation between the lenses.

Now this I can help with - I build these things!

The rule is that the ratio of the focal lengths is equal to the ratio of the beam expansion. That is, f1/f2 = d1/d2.

You can verify this through some simple ray tracing... if you sketch a collimated beam hitting a lens, it focuses at f1 away. In order to recollimate the beam, the second lens has to be positioned f2 away from this focus. The two triangles formed are similar, and hence the ratio of the beam diameters equals the ratio of the focal lengths. The two lenses are therefore a distance f1+f2 apart.

For your particular problem, the ratio of diameters is 8 - therefore the second lens has a focal length of 8*50mm = 400mm, and the distance between the two is 450mm.

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