Problem 3

Monochromatic light of wavelength

\lambda

is incident normally on a single slit of width

b

. A parallel screen is at distance

L

, with

L\gg b

so Fraunhofer diffraction applies. Intensity is observed at a small angle

\theta

from the normal through the slit centre; ignore the finite screen size and other edge effects. (a) Divide the aperture into elements and state the condition for cancellation of their amplitudes. (b) Find the angle

\theta_1

of the first dark minimum. (c) Derive its screen displacement

y_1

and explain why this is a dark fringe rather than a maximum.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

E(\theta)\propto\int_{-b/2}^{b/2}e^{ikx\sin\theta}dx

problems.proof.analysis

Add the secondary waves across the full aperture by integrating their complex amplitudes. The stated Fraunhofer and small-angle approximations are used.