## Analysis of Diffraction pattern due to single slit

** Analysis of Diffraction pattern due to single slit**: The intensity on the screen for diffraction due to single slit is

where . The intensity will be **minimum** (or zero) when

or m=1,2,...................................... or

If slit is far from the screen, is small such that where y is distance of point P from center point P_{0}

and D is distance between slit and screen.

Thus knowing m(order of minima), , D and y (separation of m'^{th} minima from principal maxima ) the slit width 'd' can be determined. Now, let us see, when we will observe **maxima** in diffraction pattern. At the center of screen, the path traveled by the secondary waves originating from one point of slit is same as that traveled by another wave on the opposite side of center of slit at the same distance and hence path difference is zero, ie all the waves meet in phase thus we observe a maxima at this point. This can also be seen mathematically for the intensity distribution equation for point , and (for small , the series expansion of will contain only first term ). Thus the intensity at the center will be

Now we will determine the position of the maxima.

or

for corresponds to minima as discussed earlier. The second condition : gives the position of maxima. Solution of transcendental equation can be determined graphically, where two functions and are plotted against **(****)**. The point where these two curves intersects, satisfies and these values of corresponds to secondary maxima in the diffraction pattern. The corresponding values of .This clearly shows that the secondary maximas are not exactly midway between the two minimas but are slightly displaced towards the principle maxima. The intensity at the mid way between two minima is where *n=1,2,3.....* and is slightly less than the intensity at the secondary maxima. The intensity of the first secondary maxima is times the intensity of principal maxima.

fig..(1)

__ Width of central (principal) maxima__ The first minima in single slit diffraction pattern occurs for . The width of central maxima at a distance D from the slit is thus given by , where is the distance between the position of peak in central maxima and first minima.

Now for equation

Width of central maxima This is the extent to which the light is distributed before diminishing first time. For a given *D*, ( ). 2y_{1} varies as . Further for a given , y is small if 'd' is large and vice versa. Also if 'd' is fixed, y is small if is small.