Diffraction FROM INTERFERENCE TO DIFFRACTION: The phenomenon of interference, is explained on the basis of superposition of two coherent light beams each obtained from two different slits and gives rise to variation in the intensity of light on the screen.

We can also get some sort of variation in the intensity of light on a screen by superposition of light waves obtained from different parts of the same slit. This phenomenon is termed as Diffraction. The phenomenon of diffraction is generally associated with the bending of light at the corners or edges of an aperture of an obstacle.

that is when an opaque object is placed in the path of light, the light encroaches in the region of geometrical shadow. So shadow is not sharp as we expect it to be. However the nature of encroachment of the light depends upon size of the obstacle.

Diffraction d>>l d = l d<<l FIG1.: As the opening size gets smaller, the wave front experiences more and more curvature

When the obstacle/opening is large compared to the wavelength, waves do not bend round the edge. When obstacle/opening is small bending round the edges is noticeable and when it is very-very small the waves spreads over entire surface behind the opening.

FIG2.: As the barrier or opening size gets smaller, the wave front experiences more and more curvature

Classification of Diffraction Depending upon the distance between source, slit and the screen we can classify diffraction into two classes: 1. Fresnel’s diffraction: In this class of diffraction both source of light and the screen are at a finite distance from the aperture or obstacle. 2. Fraunhoffer’s diffraction In this class of diffraction the distance of aperture or obstacle from the source or the screen or both is infinite. The infinite separation can be obtained by using lenses.

Frounhoffer’s diffraction at a single slit: A single slit placed between a distant light source and a screen produces a diffraction pattern. It will have a broad, intense central band The central band will be flanked by a series of narrower, less intense secondary bands Called secondary maxima The central band will also be flanked by a series of dark bands Called minima

Frounhoffer’s diffraction at a single slit (Contd.)

Let a plane wave front is incident on a slit AB. The plane wavefront may be obtained by using a collimating lens. The diffracted light is then collected on a screen with the help of a lens L. The diffracted light in the direction of incident light will come to focus at point P. All the waves arriving at P will be in phase and hence P will be a point of maximum, called central maximum.

Consider light diffracted at an angle θ Consider light diffracted at an angle θ. All light rays diffracted at angle θ, will come to focus at P1. The point P1 may be a maximum or minimum depending upon the path difference between the rays diffracting from the two extreme edges of the aperture or slit.

Let AN be the normal drawn from A on the ray diffracted at angle θ, Then the path difference between the rays diffracting from A and B will be given by BN. If the width of the slit be ‘b’, then BN = b sinθ

Suppose BN = λ and let O be the mid point of the exposed wave front Suppose BN = λ and let O be the mid point of the exposed wave front. That is O divides the slit into two equal parts. Then the path difference between the secondary waves originating from A to O will be λ/2. Corresponding to every point in the part AO of the slit, there will be a point in the part OB of the slit, such that the path difference between the secondary waves originating from them is λ/2.

the secondary wave originating from any point in part AO of the secondary wave originating from any point in part AO of the slit will interfere destructively with the secondary wave originating from the corresponding point in part OB and so the point P1 will be a minimum. Similarly if the point P1 is such that BN = 2 λ, the exposed wave front can be divided into 4 equal parts such that alternate parts produce minimum at P1.

So in general if BN = p λ, where p =1,2,3,4.... then the point P1 will be a point of minimum. If the diffraction angle for pth minimum is denoted by θp then BN = b sin θp = p λ Or sin θp = p λ/b The point P1 will be a maximum, if BN = (2p+1) λ/2.

If BN = 3 λ/2, then part of exposed wave front can be divided into three equal parts. Two adjacent parts will cancel the effect of each other, and single part left unpaired will produce illumination. As p increases the size of wave front causing illumination go on decreasing and so intensity at maximum will go on decreasing. If the distance of P1 from central maximum(at P) is yp then where f is the focal length of the lens.

WIDTH OF CENTRAL MAXIMUM Frounhoffer’s diffraction at a single slit (Contd.) WIDTH OF CENTRAL MAXIMUM For the first minimum: sin θ1 = From above equation if θ is very small then sin θ1 = θ1 We have θ1 = hence Now, sin θp = p λ/b So, θ1 = λ/b

Frounhoffer’s diffraction at a single slit (Contd.) Or The width of central maximum is 2y1, so given by (2) The width of any secondary maximum is So the width of central maximum is twice than that of any secondary maximum.

Frounhoffer’s diffraction at a single slit (Contd.)

Conclusions: The width of central maximum is proportional to wavelength and inversely proportional to the width of the slit. If b is large, y is small that is secondary minimum will be very small in size and for large slit they cannot be distinguished.

That is why diffraction pattern is not observed with wide slits. With monochromatic light the diffraction pattern consists of alternate dark and bright bands. With white light the central maximum is white and the diffraction pattern on its either side is colored.