EDAYATHANKUDY G S PILLAY ARTS AND SCIENCE COLLAGEK PREMKUMAROPTICS DIFFRACTION
DIFFRACTION AT A SINGLE SLIT • Let a parallel beam of monochromatic light of wavelength λ be incident normally upon a narrow slit of width AB=a. • The diffracted light is focused by a convex lens on a screen placed in the focal plane of the lens. The diffraction pattern obtained on the screen consist of a central bright bands of decreasing intensity on both sides. • As a planed wave front is incident on the slit AB, each point on the wavelength become source of secondary wavelets. The rays diffracted along the direction of incident rays are focused at O. The point O is optically equidistant from all points on the slit AB. Therefore, all the secondary wavelets from AB Reach O in the same phase. Hence there is maximum intensity at O.
SINGLE SLIT • Let us consider the intensity at any point P1 above O. The secondary waves travelling at an angle θwith the normal are focused at P1. Let AC be perpendicular to BC. The path difference between the secondary waves originating from extreme point A and B is. • BC= AB sinθ = a sinθ let this path difference be one wavelength. If we imagine the aperture AB to be divided into two halves, then the wavelets from corresponding points in each half will differ in phase at P1 by λ/2.
Single slit • They would mutually interfere and cancel out each other effect thus producing the first minimum at P1. • BC= a sin θ = λ • sinθ = λ/a or θ = λ/a (»» θ is very small) • Hence the first minimum on wither side O will occur in a direction given by • θ = λ/a. • Suppose for another point P2, the path difference BC is 2λ. Now the slit can be supposed to be divided into four equal parts. The rays from corresponding points separated by a distance a/4 in the two halves of each half of the slit will have a path difference of λ/2.
Single slit • These rays will mutually interfere and cancel out each other. A second minimum, therefore occurs at P2 in a direction θgiven by. BC = a sinθ = 2λ. • In general the various secondary minima will occurs when the path diffraction b/w the extreme rays is an even multiple of λ/2 or an integral multiple of λ. for minimaBC = a sin θ = 2n (λ/2) = nλ • Besides the central maximum at O, there are secondary maxima which lie in b/w the minima on either side of the central maximum. These are situated in direction in which the path difference BC is an odd multiple of λ/2. • for secondary maxima BC = a sinθ = (2n+1) λ/2………(2)
DOUBLE SLIT • Let AB and CD be two parallel slits of equal width a and separated by an opaque distance b. Let a Plano wave- front be incident normally upon the slits. The light diffracted from these slits is focused by a lens L on a screen XY placed in the focal plane of lens L. The slits and the screen are perpendicular to the plane of the paper. • By Huygens's principle, every point in the slit AB and CD sends out secondary wave lets in all direction. All the secondary waves travelling in the direction of incident light come to focus at O.
DOUBLE SLIT • From the theory of diffraction at a single slit, the resultant amplitude R due to wavelets diffracted from each slit in a direction θ is. • R= A sin α /α • Here A is a constant being equal to the amplitude due to a single slit, when θ = 0, and α = Πa sin θ/λ. • We can consider the two slits as equivalent to two coherent source placed at the middle points S1 and S2 of the slits and each sending a wavelet of amplitude A sin α/αin a direction θ to be the normal. Hence the resultant amplitude at P will be due to the interference between the two waves of same amplitude R and a phase difference φ.
PLANE TRANSMISSION DIFFRACTION GRATING • An arrangement consisting of a large number of parallel slits of equal width and separated from one another by equal opaque spaces is called a diffraction grating. • It is constructed by ruling equidistant parallel lines with a fine diamond point on an optically plane glass plate. The ruled lines are opaque to light. These are called opacities. The space in between any lines is transparent to light. The space are called the transparencies. Such a grating is called transmission grating.
DIFFRACTION GRATING • Consider a parallel beam of light incident normally on a grating XY. AB,CD,EF… are the transparent slits. Let the width of each slit be a and the width of each opaque portion be b. Then the distance (a+b) is called the Grating constant or grating element. The points in the consecutive slits separated by the distance (a+b) are called the ‘corresponding point’. Most of the light issuing from the spaces will go straight on. But as the width of space is comparable to the wavelength of light, part of this light spreads out in all direction, on leaving the slits.
DIFFRACTION GRATING • (i) Suppose a telescope with its axis normal to the grating is placed in the path of diffracted light. Then the rays issuing out normally are brought to focus at a point O lying on the principle axis of the lens L. All the rays reaching O are in phase with each other. Hence the rays reinforce producing a central bright band • (ii) The rays diffracted at an angle θwith the grating normal reach P1 on passing through the lens in different phases. Draw AK perpendicular to the direction of the diffracted light. Then CN is the path difference between the rays diffracted from the two corresponding points A and C at a angle θ.
DIFFRACTION GRATING • The path difference CN = Ac sin θ = (a+b) sin θ • If this path difference is an even multiple of λ /2,then the point p will be bright. • Hence for maximum intensity, we have • (a+b)sinθ = ± nλ • Where n is an integer , 0,1,2,3, etc. n is called the order of the interference maximum. • The point P will be dark if (a+b) sinθ = ±(2n+1)λ/2 • Thus the diffracted rays from any pair of corresponding points of the slits will produce constructive or destructive interference at a point P according as the path difference is an even or odd multiple of λ/2.
DIFFRACTION GRATING • This condition holds true for all the rays from the corresponding point of any pair of adjoining slits in the entire grating surface. We find therefore that brightness and darkness are alternate. • For n=0, we get central maximum at 0. When n=0,sinθ=0 and θ=0. • Hence when there is no diffraction, the light travels straight and is said to be of zero order • For n=± 1 sinθ1 =± λ/ (a+b)
DIFFRACTION GRATING • This gives the condition for the first order principle maximum intensity point on either side of O i.e., at P1 and P’. The intensity at P1 is less than the intensity at O. • for n=± 2, sinθ2 = ± 2λ/(a+b). • This gives the direction of the second order principle maxima. • for ±3, sinθ3 = ± 3λ/(a+b).
DIFFRACTING GRATING • Most of the light is concentrated in the principle maximum of zero order. The intensity goes on diminishing gradually as we go to the higher orders. Two orders on either side will be seen clearly with the ordinary grating we have in our laboratories. Higher order spectra will be feeble and cannot be seen clearly