Discuss the special cases on first minimum in Fraunhofer diffraction.

Explanation

Consider the condition for first minimum with (n = 1). a sin `theta = lamda`
The first minimum has an angular spread of,
`sin theta = lamda/a`
Now, we have special cases to discuss on the above condition.
(i) When `a lt lamda`, the diffraction is not possible, because `sin theta` can never be greater than 1.
(ii) When `a gt= lamda` the diffraction is possible.
(a) For `a = lamda, sin theta = 1 i.e., theta = 90^@` . That means the first minimum is at `90^@`. Hence, the central maximum spreads fully in to the geometrically shadowed region leading to bending of the diffracted light to `90^@` .
(b) For `a gt gt lamda, sin theta lt lt 1` i.e, the first minimum will fall within the width of the slit itself. The diffraction will · not be noticed at all.
(iii) When `a gt lamda` and also comparable, say `a = 2 lamda, sin theta = lamda/a = (lama)/(2 lamda) = 1/2`
` theta = 30^@` These are practical cases where diffraction could be observed effectively.
(iv) The slit AB is divided into two half's AC and CB. Now the width of AC `(a/2)`. Different points on the slit which are separated by the same width (here ` a/2` ) called corresponding points.
(v)The path difference of light waves from different corresponding points meeting at point P and interfere destructively to make it first minimum. The path difference `delta` between waves from these corresponding points is, `delta = a/2 sin theta`
(vi) The condition for P to be first minimum,
` a/2 sin theta = lamda/2`
` a sin theta = lamda ` (first minimum) .