diffraction, Fraunhofer by a narrow slit Fig. d5-a shows  cross  section of a slit of width b (and light a) on which parallel  beam  of light is incident from the left. A convex lens kept  on  the right focuses the rays emerging from the wave front on  the  screen. Consider the point P₀, situated on the axis. The secondary wavelets reaching the point from all elements of the wavefront are in phase. The resultant amplitude can be obtained by adding vectorially the amplitudes of all the individual elements In fig d5-b, the amplitude at P₀ is found by dividing the wavefront into nine elements. Let it be called A₀. Now consider a point P situated slightly above P_0. The secondary wavelets will reach P with phase difference. Let the phase difference between the secondary wavelets originating from nearest and farthest elements be d . The amplitudes from the elements when plotted will form an arc (see fig d5-c). The resultant amplitude is the vector A. The length of the arc will be however A₀. From the figure we can write,

A/A_o = BB¢ / arc BB¢ = 2 R sin (d /2) / (R d ) = sin b /b

where b is half the phase difference between the secondary wavelets from nearest and the fartest elements.

b = (1/2) ( 2p /l ) b sinq

The intensity I is proportional to the square of the amplitude,

I = I₀ sin² b /b (1)

where I₀ is the intensity at P_0.

In  fig.d6  graphs  are shown for intensity distribution and amplitude   distribution for   the   single   slit  fraunhofer diffraction.  The  maximum  intensity occurs at the P_o in fig.d5 where  all  the  secondary  wavelets arrive in phase. This point corresponds  to  b  =  0,  in the intensity distribution curve of fig.d6.  This  is called the  principal  maximum.  From the principal maximum the intensity falls to zero for all values of b given by,

b = ± np

where  n is an integer. This gives,

b sin q = ± nl

The condition for minima. In between any two successive minima,  we  observe  secondary maximum. The values of b for the secondary maxima can be obtained by differentiating eq.(1) and equating it to zero. This yields,

tan b = b

The first few values of b satisfying the above equation are

+ 1.43p , + 2.46p , 3.47p , .......

The  ratio  of  intensity  of a secondary maxima to the principal maxima,  can  be obtained by calculating sin² b / b ² , with appropriate value  of  b  given  above. The approximate intensity ratios can be calculated with value of b = 3 p /2, 5 p /2, 7 p /2 etc.