In Fig. 36.12 c the central diffraction maximum contains...

In Fig. $36.12 \mathrm{c}$ the central diffraction maximum contains exactly seven interference fringes, and in this case $d / a=4 .$ (a) What must the ratio $d / a$ be if the central maximum contains exactly five fringes? (b) In the case considered in part (a), how many fringes are contained within the first diffraction maximum on one side of the central maximum?

Key Concepts

Single?Slit Diffraction

A single slit of finite width produces a diffraction pattern whose envelope is given by the sinc^2 function. The intensity falls to zero (minima) at angles given by a sin ? = m? (with m a nonzero integer). The central maximum of this pattern extends from –?/a to ?/a in angle. This concept is crucial because it sets the overall envelope within which other interference fringes appear.

Interference Fringe Spacing

When light passes through more than one aperture, the waves interfere to produce fringes whose angular positions are given by sin ? = m?/d (where d is the separation between the slits and m is an integer order). The spacing of these fringes is directly related to the ratio ?/d. Knowing this spacing helps decide how many interference fringes can fit inside the diffraction envelope.

Modulation of Interference by the Diffraction Envelope

In many optical experiments, such as multiple-slit interference, the fine interference fringes are modulated by the broader diffraction pattern of each individual slit. The number of interference fringes seen within the central maximum is determined by the ratio d/a, as the interference maxima appear at positions that must lie within the angular span set by the diffraction minimum at sin ? = ?/a.

Missing Orders in Combined Diffraction–Interference

When an interference maximum occurs exactly at a diffraction minimum, that fringe is missing. This phenomenon explains why, even though a counting argument might suggest more interference fringes, only a smaller number are visible. For example, with d/a = 4, the interference fringes with orders m = 0, ±1, and ±2 are clearly visible, while the fringes at m = ±3 fall at the edge or beyond the diffraction envelope, yielding a total of 7 fringes in the central maximum. Similarly, adjusting d/a so that the edge fringe falls exactly at the diffraction minimum leads to a change in the number of observed fringes.

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