Diffraction

The phenomenon of bending of light and spreading into geometrical shadow after passing through obstacle or slits is called diffraction.

The diffraction pattern can be obtained on the screen when width of slit is equivalent to wavelength of light. The bright and the dark fringes obtained on the screen due to diffraction are called secondary maxima and minima.

Types of diffraction:

On the basis of arrangement of device used, there are two types of diffraction. They are:

1. Fresnel's diffraction:

The diffraction which is obtained due to the source slits and screen at a finite distance is called fresnel’s diffraction. In this diffraction, the wave front is spherical. So, diffraction pattern obtained on the screen are quite complex.

2.Fraunhoffer’s diffraction:

The diffraction which is obtained due to the source slits and screen at infinite distance is called fraunhoffer’s diffraction. In this diffraction, the wave front is plane. So, diffraction pattern obtained on the screen are simple. In this diffraction lenses are used to converge the ray of light on screen or slits.

Fraunhoffers’s single slits diffraction:

Figure shows experimental arrangement of fraunhoffers single slits diffraction. It consists of two lenses $'L_1'$, and $'L_2'$ for the converging of ray of light on slit and screen from the source ‘S’.

All parallel rays of light focus at the center of screen. So, center of screen is bright called center maxima.

On the screen, diffraction pattern i.e., bright and dark fringes is obtained due to superimpose of diffracted rays which are called secondary maxima and minima.

Draw ‘AN’ perpendicular on ‘BP’ the ‘BN’ gives path difference between ‘AP’ and ‘BP’ rays.

Therefore, path difference(x)=BN

In triangle ABN,

$\sin\theta=\frac{BN}{AB}=\frac{BN}{a}$ where a$\to$width of slits

$BN=a\sin\theta$

Therefore, path difference$(x)=a\sin\theta$….(i)

For secondary maxima,

Path difference$(x)=\frac{(2n+1)\lambda}{2}$ where n=0,1,2

$a\sin\theta=\frac{(2n+1)\lambda}{2}$

$a\sin\theta=\frac{(2n+1)\lambda}{2a}$

for small $'\theta'$, $\sin\theta\approx\theta$

$\therefore \theta_{n}=\frac{(2n+1)\lambda}{2a}$

This is the angular width of nth secondary maxima.

For n=1, $\theta_1=\frac{3\lambda}{2a}$ $\rightarrow$1^st secondary maxima

For n=2, $\theta_2=\frac{5\lambda}{2a}$ $\rightarrow$ 2^nd secondary maxima and so on.

If ‘y’ be linear width of nth fringe and ‘D’ be distance between slit and screen. Then,

$\theta=\frac{y_n}{D}$

$y_n=\theta_n D$

 

For secondary minima,

Path difference$(x)=n\lambda$

$a\sin\theta=n\lambda$

$\sin\theta=\frac{n\lambda}{a}$

for small $\theta$, $\sin\theta\approx\theta$

$\theta_n=\frac{n\lambda}{a}$   Angular width of nth secondary minima

and,

$y_n=\theta_n D$

$y_n=\frac{n\lambda D}{a}$

$y_n=\frac{n\lambda D}{a}$   Linear width of nth secondary minima

 

Width of central maxima:

The width of central maxima is twice of 1^st secondary minima

Width of central maxima=$2y=\frac{2\lambda{D}}{a}$

Angular width of central maxima=$2\theta_1=\frac{2\lambda}{a}$

 

Diffraction grating:

An arrangement of large number of parallel slits is called diffraction grating. The diffraction grating can be obtained by ruling lines over the glass surface.

When the lines are ruling over transparent glass surface then the grating is called transmission grating.

And, when lines are ruling over reflecting glass surface then the grating is called reflecting grating.

Let us consider 'N' identical slits of width ‘a’ separated by identical opaque of width ‘b’ in l inch glass then,

$Na+Nb = 1 inch$

$N(a+b) = 1$

$\Rightarrow a+b=\frac{1}{N}$

Here, $(a+b)$ is called grating element. Hence, grating element is defined as the sum of width of each slit and each opaque.

 

Path difference between rays is;

$x=(a+b)\sin\theta$

for maxima,

$(a+b)\sin\theta=n$

$d\sin\theta=n\lambda$ where $d=a+b$

 Resolving Power

The ability of an optical instrument to produce separate images of two objects very close together is called resolving power.

Rayleigh Criterion: When the central maximum in the diffraction   pattern of one point source falls over the other point source, then the two points sources are said to have been resolved by optical instrument.

Resolving power of a Microscope:

The resolving power of a microscope is defined as the reciprocal of the distance between two objects which can be just resolved when seen through the microscope.

Resolving power=$\frac{1}{\delta d}=\frac{2\mu \sin(\theta)}{\lambda}$

Resolving power depends on;

(i)wavelength, $\lambda$

(ii)refractive index of the medium between the object and the objective, and

(iii)half angle of the cone of light from one of the objects $\theta$.

 

Resolving power of a Telescope:

The resolving power of a telescope is defined as the reciprocal of the smallest angular separation between two distant objects whose images are seen separately. 

Resolving power= $\frac{1}{d\theta}=\frac{a}{1.22 \lambda}$

Resolving power depends on;

(i)wavelength $\lambda$

(ii)diameter of the objective, a