Chapter 12 Diffraction Notes

     

Unit-3 Wave and Optics Notes

Chapter 12 Diffraction Notes

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Unit-3 Wave and Optics Notes

Chapter 12 Diffraction Notes

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

Types of Diffraction

(1) Fresnel’s diffraction

The diffraction in which the source and the screen are at finite distances from the obstacle is known as Fresnel’s diffraction. In this type of diffraction, the incident, as well as the Diffraction lens, is not required

(2) Fraunhofer diffraction

The diffraction in which the source and the screen are at infinite distances from the obstacle is known as Fraunhofer diffraction In this diffraction the incident as diffraction wavefronts are plane. And in this diffraction ions are required.

Difference between interference diffraction.

Fraunhofer’s diffraction at a single slit. (Diffraction of light at a single slit.)

Fraunhofer's diffraction at a single slit ( Diffraction of light at a single slit)

Fig: Diffraction of light at a lens 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 the figure. After passing through the slit of a beam of light focused on the screen 'XY' by means of the convex lens 'G'.

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

BN = dsinθ   (∴ sinθ = BW/d)

i.e Path difference = dsinθ---(i)

nth secondary minima (dark fringes)

path difference = nλ

dsinQn = nλ

SinQn = nλ / d

If Q is very very small

SinQn ⋍ Qn

(n = 1, 2, 3, --)

for n = 1 First minima

Q1 = λ/d

for n = 2 Second minima

Q2 = 2λ/d

nth secondary maxima (bright fringes)

path difference = (n+1/2) λ

dsinQn = (2n+1) λ

SinQn = (2n+1) λ / 2d

If Qn is very very small

SinQn ⋍ Qn

Qn = (2n+1)λ / 2d

for n = 1 First minima

Q1 = 3λ/2d

for n = 2 Second minima

Q2 = 5λ/2d

Hence the condition for secondary maxima is

Qn = (2n+1) λ/2d

And that of secondary minima is

Qn = nλ/d

For n = 1, first maxima

For n = 2, Secondary maxima

Hence the condition for secondary maxima is

And that condition for secondary maxima is

Width of central maxima,

Now from the figure,

For small θn, tanθ≈ θn

Again for the nth minima

d sinθn = nλ

For small θn, sinθn ≈θn

From equations (i) and (ii)

For n = 1

For n = 2

Fringes width

Therefore the width of central maxima.

Diffraction Grafting

A diffraction grafting is a large number of fine, equidistance closely spaced parallel lines of equal width slit ruled on glass or metals. Thus, in diffraction grafting, there are a large number of extremely narrow parallel slits separately equal opaque ( not transparent space)

Theory of diffraction gravity

Fig: Theory of diffraction gravity

Let us consider a plane wavefront of monochromatic light of wavelength 'λ' for normally on a transmission grafting which is shown in the above figure

If 'N' be the number of line per inch of grafting 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 wavefront incident on a grafting surface then all the secondary waves are traveling in the same phase and converse by means of the lens at point 'p' on the screen. Hence maxima are formed on point p.

If 'Q1' be the angle of diffraction then for 1st secondary maxima.

(a + b) sinQ1 = λ

Similarly the secondary maxima,

(a + b) sinQ2 = 2λ

In general (a + b) sin Qn = 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 --- , n, 1st, 2nd, 3rd, --- nth order maxima are obtained which are much less brighter then zero-order maxima.

The intensity of variation of diffraction pattern on the screen is shown below.

Fig: Intensity distribution of diffraction gravity.

Resolving Power

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

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

Resolving Power of grafting

It is defined as the ratio of the wavelength of any spectral line to the difference in wavelength between the disc line and neighboring line.

Resolving the power of a telescope

Let D be the diameter of objective lens f ‘λ’ be the wavelength of line then angular separation ‘dθ’ is given by,

The reciprocal of angular separation is called the resolving power of the telescope.

= P ∝ D

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

Resolving the power of the microscope

The reciprocal the distance between two objects which can be just resolved. If observed through a microscope is called resolving power of the microscope.

λ = wavelength

μ = refractive index.