DIFFRACTION AT A CIRCULAR APERTURE

In our analysis of double-slit interference (Section 41-2) we assumed that the slits were arbitrarily narrow — that is, that a << l. For such narrow slits, the central part of the screen on which the light falls is uniformly illuminated by the diffracted waves from each slit. When such waves interfere, they produce interference fringes of uniform intensity.

In practice, for visible light, the condition a << l is usually not met. For such relatively wide slits, the intensity of the interference fringes formed on the screen is not uniform. Instead, the intensity of the fringes varies within an enveloped due other diffraction of a single slit.

The effect of diffraction on a double-slit interference pattern is illustrated, which compares the double-slit pattern with the diffraction pattern produced by a single slit of the same width as each of the double slits. You can see that the diffraction does indeed provide an intensity envelope for the more closely spaced double-slit interference fringes.

Let us now analyze the combined interference and diffraction pattern. The interference pattern for two infinitesimally narrow slits is given by Equation 41-13 or, with a small change in notation

in which d is the distance between the center lines of the slits.

The intensity for the diffracted wave from either slit is given by Equation 42-8 or, again with a small change in notation,

We find the combined effect by regarding I_m,int in Equation 42-13 as a variable amplitude, given in fact by I_q,diff of Equation 42-15. This assumption, for the combined pattern, leads to