Question

A collimated beam of light of diameter 2 mm is propagating along the x-axis. The beam is required to be expanded into a collimated beam of diameter 14 mm using a system of two convex lenses. If the first lens has focal length 40 mm, then the focal length of the second lens is_____ mm.

Hint:

For beam expanders, diameter ratio = focal length ratio.

Answer 1

Correct Answer: 280

To solve the problem of expanding a collimated beam using two lenses, we use the concept of a beam expander consisting of two lenses. The beam expander uses the principle of a telescope system where the magnification \(M\) is given by the ratio of the diameters of the output beam to the input beam. Here, \(M = \frac{D_2}{D_1}\), where \(D_2 = 14 \text{ mm}\) and \(D_1 = 2 \text{ mm}\).

First, calculate the magnification:
\(M = \frac{D_2}{D_1} = \frac{14}{2} = 7\).

The magnification in a two-lens system is also given by the ratio of the focal lengths of the second lens (\(f_2\)) to the first lens (\(f_1\)): \(M = \frac{f_2}{f_1}\). Substituting the known magnification and the focal length of the first lens, we have:
\(7 = \frac{f_2}{40}\).

Solve for \(f_2\):
\(f_2 = 7 \times 40 = 280 \text{ mm}\).

This calculation gives \(f_2 = 280 \text{ mm}\), which is within the specified range (280,280). Thus, the focal length of the second lens is confirmed to be correct and fits the expected range.