The beam from an argon laser has a diameter of 3.00 mm and a power output of 5.21 W. The beam is focussed onto a diffuse surface by a lens of focal length . A diffraction pattern is formed. (a) We have to show that the radius of the central disk is given by

.

The central disk can be shown to contain 84% of the incident power. We have to calculate (b) the radius of the central disk, and the average power flux density (c) in the incident beam and (d) in the central disk.

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(a)

A result of Fraunhofer diffraction is that image formed of a distant object by a lens at its focal point is not a point but a circular disk surrounded by several progressively fainter secondary rings. The first minimum occurs at an angle from the central axis given by

,

where is the diameter of the aperture. As , . As the diffraction ring is formed at the focal plane of the lens, its radius will be

.

(b)

We calculate the radius of the central disk using the data of the problem:

We find

(c)

The average power flux density in the incident beam,

(d)

It is given that the central disk receives about 84% of the incident power of the beam. The power of the central disk will therefore be

Therefore, the average power flux density in the central disk will be