Class 12 Physics Diffraction Notes

Class 12 Physics Diffraction Notes
UNIT 3
CLASS 12 PHYSICS • WAVE AND OPTICS

Diffraction

Chapter 12
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Diffraction

The bending of light from the corner of an obstacle and the splitting into the region of geometrical shadow is known as diffraction. The diffraction of light produces dark and bright fringes of unequal width known as diffraction fringes.

Types of Diffraction

1. Fresnel’s Diffraction

The diffraction in which the source and the screen are at finite distance from the obstacle is known as Fresnel’s diffraction. In this type of diffraction, the incident as well as diffracted wave fronts are spherical or cylindrical. In this type of diffraction lens is not required.

2. Fraunhofer Diffraction

The diffraction in which source and screen are at infinite distance from the obstacle is known as Fraunhofer diffraction. In this diffraction the incident as well as diffracted wave fronts are plane. And in this diffraction lens is required.
Conceptual comparison of Fresnel and Fraunhofer diffraction Fresnel diffraction Source Screen Fraunhofer diffraction Screen
Fresnel and Fraunhofer diffraction

Difference Between Interference and Diffraction

Interference Diffraction
This is formed due to superposition of wavelets from different wave fronts. This is formed due to superposition of wavelets from same wave front.
All the bright and dark fringes are of equal width. All the bright and dark fringes are not of equal width.
The intensity of all bright and dark fringes are same. The intensity of all bright and dark fringes are not same.
Points of minimum intensity are perfectly dark. Points of minimum intensity are not perfectly dark.
Path difference for nth maxima is nλ. Path difference for nth secondary maxima is (n + 1/2)λ.
Path difference for nth minima is (n + 1/2)λ. Path difference for nth secondary minima is nλ.

Fraunhofer’s Diffraction at a Single Slit

(Diffraction of light at a single slit)

Fraunhofer diffraction at a single slit with lenses, slit AB, observation point P, angle theta and path difference construction Source S Lens L₁ Lens L₂ d A B θ P Screen XY
Fraunhofer diffraction of light at a single slit

Let us consider a parallel beam of light incident normally on a slit ‘AB’ of width ‘d’ from source ‘S’ through lens L1 as shown in figure. After passing through slit, beam of light is focused on the screen ‘XY’ by means of convex lens L2.

Suppose a point ‘P’ on the screen at which light wave is travelling in a direction making angle ‘θ’ with ‘CD’. The wavelets from different part of slit will not reach at point ‘P’ in same phase. As a result they cover unequal distance to reach ‘P’ and path difference of light wave reaching at point ‘P’ from A and B is given by:

BN = d sinθ
Path difference = d sinθ   — (i)

nth Secondary Minima (Dark Fringes)

Path difference = nλ
d sinθn = nλ
sinθn = nλ/d

If θ is very very small:

sinθn ≈ θn
θn = nλ/d,   n = 1, 2, 3 …

For n = 1, first minima:

θ1 = λ/d

For n = 2, second minima:

θ2 = 2λ/d

nth Secondary Maxima (Bright Fringes)

Path difference = (n + 1/2)λ
d sinθn = (2n + 1)λ/2
sinθn = (2n + 1)λ/(2d)

If θn is very very small:

sinθn ≈ θn
θn = (2n + 1)λ/(2d)

For n = 1, first maxima:

θ1 = 3λ/(2d)

For n = 2, second maxima:

θ2 = 5λ/(2d)
Condition for secondary maxima:
θn = (2n + 1)λ/(2d)
Condition for secondary minima:
θn = nλ/d

Width of Central Maxima

Central diffraction maximum bounded by first minima at plus and minus theta one θ₁ θ₁ D y₁ y₁ Central maxima
Geometry used to obtain width of the central maximum

Now from figure:

tanθn = yn/D

For small θn, tanθn ≈ θn:

θn = yn/D   — (i)

Again for nth minima:

d sinθn = nλ

For small θn, sinθn ≈ θn:

θn = nλ/d   — (ii)

From equations (i) and (ii):

yn/D = nλ/d
yn = nλD/d

For n = 1:

y1 = λD/d

For n = 2:

y2 = 2λD/d

Fringe width:

y2 − y1 = λD/d

Therefore the width of central maxima:

Width of central maxima = 2λD/d

Diffraction Grating

A diffraction grating is a large number of fine, equi-distance, closely spaced parallel lines of equal width slit ruled on glass or metals. Thus in a diffraction grating there are large number of extremely narrow parallel slit separating by equal opaque (not transparent) space.

Theory of Diffraction Grating

Transmission diffraction grating with multiple slits, lens and screen forming central and higher order maxima Grating Lens Screen P′ P
Theory of diffraction grating

Let us consider a plane wave front of monochromatic light of wavelength ‘λ’ incident normally on a transmission grating as shown in above figure.

If ‘N’ be the number of line per inch of grating, then:

a + b = 1/N inch
a + b = 2.54/N cm   — (i)
where a = width of each slit
b = width of opaque portion

If a plane wave front incident on a grating surface then all the secondary wave are travelling in same phase and converge by means of lens at point ‘P’ on the screen. Hence maxima is formed on point P.

If θ1 be the angle of diffraction, then for 1st secondary maxima:

(a + b) sinθ1 = λ

Similarly for second secondary maxima:

(a + b) sinθ2 = 2λ

In general:

(a + b) sinθn = nλ
where n = 1, 2, 3 …

For n = 0, the central maxima is formed. It is also called zero order maxima.

For n = 1, 2, 3 … first, second, third … nth order maxima are obtained which are much less brighter than zero order maxima.

Intensity Distribution of Diffraction Grating

Intensity distribution with a strong central maximum and weaker side maxima Central maxima −θ +θ
Intensity distribution of diffraction pattern

Resolving Power

It is defined as the ability of an optical system to form separate image of an object that are very close to each other.

A resolving power depends on the wavelength of light used and the size of slit.

Resolving Power of Grating

It is defined as the ratio of wavelength of any spectral line to the difference in wavelength between this line and neighbouring line.
Resolving power = λ/Δλ

Resolving Power of Telescope

Let D be the diameter of objective lens and ‘λ’ be the wavelength of light. Then angular separation ‘dθ’ is given by:

dθ = 1.22λ/D

The reciprocal of angular separation is called resolving power of telescope:

P = 1/dθ
P = D/(1.22λ)
P ∝ D

This relation shows that resolving power of telescope can be increased by increasing the diameter of objective lens.

Resolving Power of Microscope

The reciprocal of distance between two objects which can be just resolved if observed through microscope is called resolving power of microscope.
P = 1/d
But, d = λ/(2μ sinθ)
P = 2μ sinθ/λ
where λ = wavelength
μ = refractive index
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