10.5.4 Diffraction Envelope

Now let’s look at the photo of the double-slit interference pattern again. Do you notice that the bright fringes are not equally bright? Do you notice that only the low order bright fringes are visible?

In the past, we have treated a double-slit as two point sources. But this is not true. A double-slit consists of two slits which have non-zero widths. A double-slit is not just a double-slit, but also two single-slits at the same time.

This means that when light is shone through the double-slit, two interferences are happening at the same time: (1) the rays from one slit interfere with rays from the other slit, (2) rays from the same slit interfere among themselves.

As a result, the slit separation d determines the positions of the bright fringes, since d\sin \theta =n\lambda determines the outcome of the superposition of the two light beams from the two slits at each θ. However, the slit width b determines the brightness of the fringes at different θ,since b\sin \theta determines the brightness of each light beam at different θ (even before they superpose with each other).

As a result, we get a double-slit interference pattern whose brightness is modulated by a single slit interference pattern. Visually, it looks like a double-slit pattern inscribed in a single-slit diffraction envelope.

In fact, very often, only the (double-slit) fringes formed within the (single slit’s) central maximum are bright enough to be visible. The higher order (double-slit) fringes formed beyond the (single slit’s) first minimum angle q1 are too dim to be seen. This is simply because at those angles, the light rays from the same slit destroy themselves completely or near completely.

Applet

N-slit Diffraction (northwestern.edu)

Concept Test

2046

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