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-

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

Diffraction at N-slits of Grating:

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 = A0 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 (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-

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 (A0 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 = (A0 sin α/α) (sin Nβ/sin β) ……….(vi)

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

I ∝ A2
or I = (A02 sin2 α/α2) (sin2 Nβ/sin2 β)
I = I0 (sin α/α)2 (sin2 Nβ/sin2 β) ……….(vii)

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.

Conditions of Maxima and Minima:

Case I- Principal Maxima:

The resultant intensity at P’ is given by-
I = I0 (sin α/α)2 (sin2 Nβ/sin2 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 1st principal maximum.
n = 2, 2nd 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.
Conditions of Maxima and Minima for Diffraction Grating

Distinguish Between Cyclotron and BetatronVapour Pressure and Raoult’s Law
Fraunhofer Diffraction at a Single SlitIdeal and Non-Ideal Solutions
Diffraction PhenomenonColligative Properties
Difference Between Interference and Diffraction of LightIonic Equilibrium– NIOS

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