For constructive interference, the phase difference between the two interfering waves is

Explanation

Suppose two narrow, closely- spaced, parallel slits `S_(1) and S_(2)` of equal widths with slit separation d are illuminated by monochromatic light of wavelength ` lambda`. They serve as coherent sources. The interference pattern is observed on a screen placed parallel to the slits and a distance D from the slits `(D gt gt d)`.
For a point P on the screen, equidistant from `S_(2), and S_(2)`, the path difference `Delta = 0`. Hence, point P will be bright, the central bright fringe or fringe. For a point Q on the screen at a distance x from `(x lt lt D)`, the path difference is
` Delta = (xd)/D` ...(1)
Point Q will be bright ( maximum intensity ) if `Delta = n lambda`, where n = 0, 1, 2, 3, .... . Point Q will be dark ( minimum intensity , equal to zero ) if `Delta = (2m - 1) lambda/2`, where m = 1, 2, 3 , ... .
Thus , the interference consists of alternate bright and dark fringes on both sides of the central bright fringe .
Let `x_(n) and x_(n+1)` be the distances of the nth and (n +1) th bright fringes from the central bright fringe.
` :. (x_(n)d)/D = n lambda ` or
` x_(n) = (n lambda D)/d ` ...(2)
and `(x_(n+1)d)/D = n lambda or x_(n+1) = ((n+1)lambdaD)/d ` ...(3)
Therefore , the distance between consecutive bright fringes
` = x_(n+1) - x_(n) = (lambdaD)/d [(n+1)-n] = (lambdaD)/d` ...(4)
Let `x_(m) and x_(m+1) ` be the distances of the mth and (m+1) th dark fringes from the central bright fringe.
` :. (x_(m)d)/D = (2m - 1) lambda/2 and (x_(m+1)d)/D = [2 (m+1)-1] lambda/2 = (2 m +1) lambda/2` ...(5)
` :. x_(m) = (2m -1) (lambdaD)/(2d) and x_(m+1) = (2m +1) (lambdaD)/(2d)` ...(6)
Therefore, the distance between consecutive dark fringes
` = x_(m+1) - x_(m) = (lambdaD)/(2d)[(2m +1) -(2m-1)] = (lambdaD)/d` ...(7)
Thus , the distance between consecutive bright fringes equals the distance between consecutive dark fringes, i.e., the bright and dark fringes are equally spaced. The fringe width or band width, ` X = (lambdaD)/d`.