## 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-

d = (1/15,000) inch d = (2.54/15,000) cm |

### 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-

A = A_{0} sin α/α ……….(i)where, α = (π/λ) d sin θ ……….(ii) |

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 (A_{0} sin α/α).

Let there be S_{1}, S_{2}, S_{3}, ………. S_{N} 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-

Path Difference = (a + b) sin θ ……….(iii) and ∴ Phase Difference = (2π/λ) (a + b) sin θ ……….(iv) |

Hence to find out the resultant amplitude of P’, we have to find out the resultant of N-vibrations of equal amplitude (A_{0} sin α/α), equal periods and constant phase difference (2π/λ) (a + b) sin θ.

Let (2π/λ) (a + b) sin θ = 2β, say ……….(v) |

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

A = (A_{0} sin α/α) (sin Nβ/sin β) ……….(vi) |

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

I ∝ A^{2} or I = (A _{0}^{2} sin^{2} α/α^{2}) (sin^{2} Nβ/sin^{2} β) I = I _{0} (sin α/α)^{2} (sin^{2} Nβ/sin^{2} β) ……….(vii) |

Here the factor I_{0} (sin α/α)^{2} gives intensity distribution in diffraction pattern due to a single slit and (sin^{2} Nβ/sin^{2} β) gives intensity distribution due to N-slits.

### Conditions of Maxima and Minima:

Case I- Principal Maxima:The resultant intensity at P’ is given by- I = I _{0} (sin α/α)^{2} (sin^{2} Nβ/sin^{2} β)Intensity is maximum when sin β = 0 ⇒ β = ± nπ i.e. (π/λ) (a + b) sin θ = ± nπ or (a + b) sin θ = ± nλ ……….(viii) The above equation is also called as “grating law” in which n = 0, 1, 2, 3, —— is an order of diffraction. And for n = 0, the position is called the central principal maximum. n = 1, it is called 1 ^{st} principal maximum.n = 2, 2 ^{nd} principal maximum and so on. |

Case II- Minima:Intensity is minimum when sin Nβ = 0, but sin β ≠ 0. ∴ Nβ = ± mπ, where, β = (π/λ) (a + b) sin θ i.e. β = (π/λ) (a + b) sin θ = ± mπ/N or (a + b) sin θ = ± mλ/N ……….(ix) Where ‘m’ can have all integral values except 0, N, 2N, …….., nN, because for these values we get principal maxima. |

Case III- Secondary Maxima:In between two minima, which are located between two adjacent principal maxima, the secondary maxima are observed. |