q is the angular position of the first bright fringe.

Rays from a and b to a given point on the screen are very nearly parallel to each other. In this analysis we will assume that they are parallel. Assume that these two rays meet at a point on the screen corresponding to the first bright fringe. Then the path difference is equal to l.

From the enlarged view of the slits it is clear that this path difference is equal to dsinq.

Other bright fringes will occur for path differences of 2l, 3l etc.

Therefore, for a bright fringe to be seen

On the next diagram, the broken lines are lines from the centre of the double slit to the maxima of the interference pattern. Even though the diagram is not to scale, it is clear that the angles (q) are very small angles (much smaller than on this diagram).

As shown above, ................nl = dsinq

If we consider the first maximum, then n = 1 and as the angles are small, sinq = q in radians (which equals s/D).

Therefore, we have

and so the fringe spacing is given by

.......
 

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