Fresnel half-period zone I am following this set of lecture notes (see below). I understand  that the idea is to evaluate the Fresnel integral for the circular aperture using the graphic method. However I don't understand at all what the lecture notes are tying to tell me. Could someone please either point me to a good explanation of the phenomena or preferably paraphrase the content of those slides? 
Thank you



 A: Very simply:
When you want to determine the final intensity of a diffraction pattern at a particular point P, you need to sum the contributions of light from every possible point Q in the aperture at that point. This means that for each little bit of the aperture you need to compute the distance PQ, and in particular the phase shift (given by $2\pi$ times distance divided by wavelength). 
Now adding all those infinitesimal contributions takes a complicated integral. Luckily, for circularly symmetrical apertures where P is on the axis of symmetry, you can take the integral one annulus at a time (all points at the same distance off axis are the same distance from P). Each of these annuli adds an intensity contribution with increasing phase shift - so adding them all together leads to the spiral.
Now if you only allow light through those areas of the circular aperture which contribute to the part of the spiral that goes up (positive Y), then you can increase the intensity at the spot by a lot - this is the concept behind the Fresnel half-zone plate. This is illustrated in the following diagram (adapted from figure 136 in your presentation):

As an interesting side note, if you could somehow change the phase for each annulus so all components end up with the same phase, then you would stretch out the spiral into a straight line which would be even more efficient. And this is exactly what a conventional convex lens does... I hope that additional insight helps, rather than confuses!
ADDENDUM
You asked explicitly about Poisson's spot. To understand how that works, we look again at the spiral diagram (the sum of all those infinitesimal annuli). If you take the integral all the way to infinity, you end up at the center of the spiral. If you leave out some small part of the integral, say the first couple of rotations, the amount you are left with is the vector from the center of the spiral ("all the aperture") to the point on the outer part of the spiral ("the integral for the first bit") - and because the spiral becomes smaller only slowly, the magnitude of that vector is almost independent of the size of the obstacle (as long as the obstacle is small). This is why a bright spot is visible on axis for a circular obstacle.
There is a fun anecdote associated with this. When Poisson first heard of the theory of light was a wave, he was outraged - he thought it was the stupidest thing he had ever heard. Being a very clever man, he proceeded to "destroy" the arguments of the wave theory people by demonstrating that if this theory were actually true, it would have to result in a bright spot on axis right behind a circular obstacle. Hahahahaha - this was so obviously ridiculous and wrong that the wave theory would be sent back under the stone it came from. And then somebody did the experiment and the spot was there... The funny thing is that it was named for Poisson (who was forever reminded of this "blunder") and not for the person who actually did the experiment - Dominique Arago. See for example Poisson's spot: the greatest burn in physics
