How much diffraction spreading does a light beam undergo

Chapter 38, Problem 72

(choose chapter or problem)

How much diffraction spreading does a light beam undergo? One quantitative answer is the full width at half maximum of the central maximum of the single-slit Fraunhofer diffraction pattern. You can evaluate this angle of spreading in this problem. (a) In Equation 38.2, define f 5 pa sin u/l and show that at the point where I 5 0.5Imax we must have f 5 !2 sin f. (b) Let y1 5 sin f and y 2 5 f/!2. Plot y1 and y 2 on the same set of axes over a range from f 5 1 rad to f 5 p/2 rad. Determine f from the point of intersection of the two curves. (c) Then show that if the fraction l/a is not large, the angular full width at half maximum of the central diffraction maximum is u 5 0.885l/a. (d) What If? Another method to solve the transcendental equation f 5 !2 sin f in part (a) is to guess a first value of f, use a computer or calculator to see how nearly it fits, and continue to update your estimate until the equation balances. How many steps (iterations) does this process take?