# Theory of Diffraction Grating

## Theory of Diffraction Grating:

### Plane Diffraction Grating:

An arrangement consisting of a large number of parallel slits of the same width and separated by equal opaque spaces is known as a diffraction grating.

Gratings are constructed by ruling equidistance parallel lines on a transparent plate such as a glass plate, with a fine diamond point. The ruled lines are opaque to light while the space between any two lines is transparent to light and it acts as a slit.

### Grating Element (d):

If ‘a’ is the width of each slit and ‘b’ is the width of opaque space, then d = (a + b) is called the grating element which is nothing but the centre-to-centre spacing between any two successive slits.

Example- If there are 15,000 lines ruled per inch (i.e. 2.54 cm), then the grating element is given by-

### Diffraction at N-slits of Grating:

We know that the resultant amplitude of light diffracted at a single slit through angle ‘θ’ is given by-

Let us now consider that there are total ‘N’ parallel slits of grating, then all the secondary wavelets in each slit can be replaced by a single wave of amplitude (A0 sin α/α).

Let there be S1, S2, S3, ………. SN mid-points of N parallel slits. We want to find out the resultant effect due to N-vibrations of N-slits. For N-slits we can take use of an amplitude-phase diagram.

If ‘θ’ is the angle of diffraction, then the path difference between rays diffracted from two adjacent slits is-

Hence to find out the resultant amplitude of P’, we have to find out the resultant of N-vibrations of equal amplitude (A0 sin α/α), equal periods and constant phase difference (2π/λ) (a + b) sin θ.

Then by vector addition method, the resultant amplitude of N-waves is given by-

and the corresponding resultant intensity at P’ is given by-

Here the factor I0 (sin α/α)2 gives intensity distribution in diffraction pattern due to a single slit and (sin2 Nβ/sin2 β) gives intensity distribution due to N-slits.