Theory of Interference Fringes

Let S be a narrow slit illuminated by monochromatic light of wavelength ?, and S ₁ and S ₂ be the two narrow slits separated by a small distance 2 d (as in Figure 5). Let a screen XY be placed at a distance D from the coherent sources S ₁ and S ₂. Let O be the foot of the perpendicular drawn from C, the mid point of S ₁ and S ₂ on the screen. Now consider a point P on the screen at a distance x from O at which the condition for bright and dark fringes are to be determined. Draw S ₁M and S ₂N perpendiculars from S ₁ and S ₂ on the screen and join S ₁P and S ₂P. The path difference between the waves reaching at P from S ₁ and S ₂ is given by

In a right angled triangle S ₂NP, we have

Figure 5

Using the binomial theorem and neglecting higher terms, since D >> x or d, we get

(1)

Similarly in a right angled triangle S ₁MP, we have

(2)

? Path difference,

(3)

If the sources S ₁ and S ₂ are in the same phase as the waves reaching at P, the path differences must only be due to path difference, ?