diffraction, Fresnel class To find the effect of interference of secondary wavelets arising out of an wavefront of any shape falling on the screen kept at a finite distance, Fresnel observed that the amplitude A, of the secondary wavelets arising out of a wavefront should not be the same in all direction. It should be maximum in the forward direction and zero in the backward direction. For the quantitative variation of amplitude with direction, he introduced a factor equal to (1 + cos q ), called the obliquity factor, where q is the angle with the forward direction. The amplitude of the secondary wavelets must be proportional to the area from where they originate (ds), and it should be inversely proportional to the distance d, between the wavefront and the point on the screen.
A = constt. [ds/d]( 1 + cos q )
Now imagine a plane wavefront for simplicity and consider a point P at a distance b in front of the wavefront (see fig.d7). Imagine spheres with radii, b + l /2, b + 2(l /2) , b + 3( l /2) ..... with P as center. These will cut circles on the wavefront. The central circle is called the first half period zone. Subsequent half period zones are annular areas between the circles. It can be easily shown that all the half period zones have the same area. The significance of these zones is that the secondary wavelets arising out of a particular zone are all of the same phase, while the adjoining ones are opposite in phase. The amplitude due to the hole wavefront is the sum of the series,
A = A1 - A2 + A3 - ............ + (-1)m-1 Am (1)
The successive amplitudes decrease gradually because of obliquity factor and distance. We may write
A2 = (A1 + A3 )/2; A4 =(A3 + A5)/2 and so on
With this approximation eq. (1) reduces to
A = A1 /2+ Am/2 for m odd,
and A = A1 /2- Am/2 for m even.
For m very large Am will be very small, and
A » A1/2
That is the amplitude due to the whole wavefront is only half the amplitude produced by the first half period zone.
The diffraction by a circular aperture or by a circular obstacle can easily be explained by the concept of half period zones. Consider a circular aperture of radius r in front of a screen. When the slit is illuminated concentric circular fringes are produced. If we gradually reduce the distance between the slit and screen, the fringe pattern will shrink and the center of the pattern will alternately become bright and dark. This happens because as the distance is reduced the number of half period zones also reduce. Since the alternate terms of the expression (1) are of opposite sign the center will be alternately of high and low intensity. Ultimately only one half period zone remains, when a bright spot is produced at the center. It is easy to see that for circular obstacle the center of the fringe pattern will always be bright.
Fresnel diffraction with straight edge and cylindrical wavefront : The cylindrical wavefront generated by a parallel beam and rectangular slit, can also be divided into half period zones as done in plane wavefront. The half period zones will have the shape of strips. In this case however the areas of the half period zones are not equal. The area of half period zones decrease rapidly as we go from lower to higher half period zones. If we divide the strips further into equal parts, and construct the amplitude diagram, by placing the amplitudes of successive strips, as shown in the fig.d8, the result is a spiral. This is because the successive strips have decreasing amplitude and the phase also changes gradually. If the strips are made of infinitesimal width we get the smooth curve (fig.d9), which is called the Cornu's Spiral. Z and Z' are the eyes of the spiral. A vector joining Z and Z' gives the resultant amplitude of the whole wavefront.
The application of Cornu's spiral can be demonstrated in case of diffraction due to straight edge as shown in fig.d10. We start with the point P, which is at the edge of the geometrical shadow. At this point the amplitude is given by OZ (Fig.10), which is half of ZZ'. Therefore at P the intensity is one fourth of the unobstructed wavefront. As we move upwards, the tail of the amplitude vector moves to the left along the spiral while its head remains fixed at the eye Z. The tail will go through maxima and minima alternately, till we are close to Z', where the amplitude becomes almost constant. If we move down into the region of geometrical shadow, the tail of the amplitude vector moves up and length of the amplitude vector decrease steadily approaching zero. The corresponding intensity curve is shown in fig. d11.