EXERCISE 3: THE KNIFE-EDGE PROFILE

You can experimentally determine the value of the parameter  that describes the width of the beam by measuring the profile of the laser beam using a knife-edge mounted on a linear translation stage together with an optical detector connected to a power meter. The idea is that you allow the laser beam to enter the detector while you translate the knife-edge across the beam and monitor the power meter reading. Initially, when the beam is not blocked, the reading gives the total power of the beam; finally, the beam is completely blocked from entering the detector by the knife-edge and the reading would be zero (in the absence of background light). A schematic is shown below; we assume the beam is centered on the origin.

Imagine that the knife-edge travels in the -direction, and that the position of the knife-edge is . Then the power received by the detector is

where  is the total power of the laser beam. Show that this expression is equivalent to

where  is the error function. Calculate  and plot this quantity versus , the position of the knife edge, when  mm and for .