Cantenna and waveguide I'm studying for a lecture of RF and wave guides and i'm currently building a cantenna. I have two questions about it construction and on how it works.
My questions are about the lengths of the active element, $\lambda/4$ and the minimum can length, $(3/4)\lambda$. Why is it $\lambda/4$ and why the can can be $(3/4) \lambda$ and doesn't have to be 1 wavelength, in order to catch the "complete wave".
 A: Simplistically, the back of the cantenna acts as a reflector: if you are exactly $\frac14\lambda$ from the end, your active element is at a point where the electric field is strongest (an antinode) which is a good thing. But that could be achieved (in principle) by just placing a reflector at that distance behind the active element - so what is the rest of the device doing?
The gain of an antenna is directly related to its directionality; this in turn is most easily understood by looking at it as a transmitter. The reciprocity theorem tells us that the gain is the same for transmitting and receiving - so if we can figure out the transmission pattern, we understand the receive gain.
If you have a circular aperture that is transmitting a signal, the forward "peak" can be estimated by looking at the position of the first null: like for optics, that happens at an angle of 
$$\theta = \frac{1.22\lambda}{d}$$
To make the angle small (the antenna more directional, i.e. higher gain) you want $\theta$ small, so $d$ large. However, there is another thing we worry about - that is the behavior of the wave inside the waveguide.
Waveguides have different modes: the antenna works best when the only propagation mode is the TE11 mode (which has radial electric fields - the one you want). If you allow higher modes, you will spread the energy between the different modes and this reduces the efficiency of the antenna. Now every mode has a cutoff frequency which is related to the diameter of the waveguide: this sets an upper limit on the diameter of the tube that we want. The mode we want to avoid is the TM01 mode, which has a cutoff frequency $\omega_c = 2.4\frac{c}{b}$ (see these lecture notes, figure 2). Unfortunately this tells you that you can't make your cantenna diameter as large as you would like: this in turn sets an upper limit on the gain you can achieve.
[One way around this problem is to use a tapered waveguide - a microwave horn that has a larger aperture, and therefore greater directionality, but that tapers to a diameter that cuts out the modes you do not want. Best of both worlds. But not really a cantenna any more - but see this link for instructions on building a cantenna with fluted aperture.]
Now for the $3\lambda/4$ thing: 
If the aperture is to be an efficient "source", it needs to have a strong signal - you would like an antinode at this place. This could in principle be achieved with 1/4, 3/4, 5/4 etc wavelengths; but 1/4 is too short (your active element would be right at the edge of the can, and the standing wave would not be fully formed); using values beyond 3/4 will not improve performance, but will result in greater losses: currents are generated in the walls of the can, and these cause losses. You want the can "long enough to create the standing wave, and short enough not to contribute to losses".
Note, by the way, that the wavelength in a waveguide is not the same as the wavelength in free space; it depends on the mode, and the dimensions of the waveguide. Close to the critical (cutoff) frequency the wavelength goes to infinity... in general it is longer in the waveguide than outside, but you do need to know the size of the waveguide and the wavelength of interest in order to get the dimensions "just right". It helps that you are trying to be "near the maximum" so it's OK if you are a little bit off. And the active element is usually only 1/4 of a wavelength - the standard monopole.
And that's why you end up with the dimensions you see...
