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It is commonly known that electromagnetic radiation from an antenna is polarized; the electric field is parallel to the rod and the magnetic component is perpendicular to it.
What does the radiation of a magnetron look like? Is this radiation polarized, is it continuously or periodically declining and increasing? Is it nearly monochromatic or does it consist of many frequencies?
Paraphrasing the Wikipedia entry for the magnetron, we learn that,
... a stream of electrons passes by the openings to metal cavity resonators and causes microwaves to oscillate within, similar to the way a whistle produces a tone when excited by an air stream blown past its opening. The resonant frequency is determined by the cavities' physical dimensions.
A magnetron produces microwave energy, but it is not meant to radiate that energy. The microwaves which oscillate within the cavity resonators are not polarized in the sense that an antenna can be horizontally, vertically, circularly, co-polarized or cross-polarized. Rather, they correspond to particular field modes.
This page on magnetron coupling methods informs us that the energy is coupled from the magnetron to the load - which could be an antenna - via an E-field loop (A), an H-field loop (C), or an aperture (E), as shown below:
I'm sorry if this answer doesn't answer your literal questions – it's just that these questions aren't strictly answerable.
it is commonly known that electromagnetic radiation from an antenna is polarized; the electric field is parallel to the rod and the magnetic component is perpendicular to it.
It is commonly known that antennas come in more shapes than rods, and can have all kinds of polarizations, linear or circular or anything in between.
The magnetron is a microwave resonator that feeds into a waveguide. That waveguide's geometry then defines the modes that propagate in the waveguide. (Modes are a bit more complicated than just "this is a linear polarization")
If you want the energy from that waveguide to be radiated, you need to convert the waveguide into an antenna – for example, attach a specifically shaped horn, or helical cavity, or … That feed will then define the polarization of the free-space wave.
What does the radiation of a magnetron look like?
warm!
Is this radiation polarized,
Depends on the waveguide, but for guided waves, as said before, they have a mode, not just a polarization.
is it continuously or periodically declining and increasing?
Don't know what that means, but every wave's field amplitudes are periodically decreasing/increasing, that's what makes it a wave.
Is it nearly monochromatic or does it consist of many frequencies?
A magnetoron is a resonator, and thus produces mainly one tone.
I once wondered if a microwave oven magnetron might make a good transmitter, for CW for EME.
I took a spectrum analyser and antenna to the kitchen and microwaved a glass of water, and I saw...
The most awful collection of strong tones, covering a large part of 2.4 to 2.5 GHz, all of them shifting up and down in frequency, rising and falling, as the water rotated.
I think the magnetron was strongly influenced by the food cavity, and the very many fast-changing modes supported there, depending on the position of the absorber. (imagine measuring the return loss of a microwave oven cavity, what a mess it would be. Remember a glass of water in a microwave oven is hardly a 1:1 SWR - it's still a terrible match and there are enormous standing waves). I've no idea what the spectrum would look like into a perfect load.
I didn't get a picture, this was before phone cameras, but I found a video that looks similar, here.
Magnetron for radio transmission, as opposed to heating, might be different. At least, with the right external cavity, they can probably be cleaned up a lot compared to the one I saw. The magnetron alone is not particularly high Q, a microwave oven one is probably happy to "whistle" at any frequency between 2.4 and 2.5 GHz, depending on the external circuit.
As for polarisation - remember the power from a magnetron is coupled out by a small slot or loop, into the waveguide. So there is no "magnetron polarisation", any more than there is "transistor polarisation" - its entirely due to the following components. If it goes into a simple rectangular waveguide and then a pyramidal horn, it will be perfectly linear, but that's just one choice.
