10.5.2 Derivation of the bsinθ Formula (Beyond the Syllabus)

The single slit pattern is the result of superposition of an infinite number of rays propagating from an infinite number of points along the slit width. The outcome obviously depends on the phase difference among the rays.

Notice that the path difference between the two rays from the edge of the slit (labelled as A and C in the diagrams) is $\delta =b\sin \theta$ , b being the slit width. Consider the case when $b\sin \theta =\lambda$ . You may be thinking of C.I. occurring between A and C since their path difference is l. Or you may be thinking of D.I. occurring between A and B since their path difference is $\displaystyle \frac{\lambda }{2}$ . And what about all the other rays? You may be thinking, “what a mess”.

Surprisingly, there is a simple way to sort out this mess. In our minds, we can split the slit into two halves. The top half sends out rays A₁ to A_N while the second half send out rays B₁ to B_N. If A₁ and B₁ are in anti-phase with each other, so are A₂ and B₂, A₃ and B₃, and every pair of rays until A_N and B_N. Pairing up the rays this way makes it clear that all the rays will superpose to zero when A and B are in antiphase, which occurs when A and C are in-phase. So we conclude that a complete D.I. occurs when $b\sin \theta =\lambda$ .

If we now mentally divide the single slit into 2 slits of width $\displaystyle \frac{b}{2}$ each, it is easy to realize that the next complete D.I. occurs when $\displaystyle \frac{b}{2}\sin \theta =\lambda$ .

We can also mentally divide the single slit into 3 slits of width $\displaystyle \frac{b}{3}$ each, and realize that complete D.I. also occurs when $\displaystyle \frac{b}{3}\sin \theta =\lambda$ .

In fact, there is nothing to stop us from mentally dividing the single slit of width b into n number of slits of width $\displaystyle \frac{b}{n}$ , and conclude that complete D.I. occurs when