How beam focusing looks like in electron microscope? I mean I know there are electrostatic/electromagnetic lenses which does focus the beam, but I am not sure how it is possible to foсus beam down to a few 10nm while emitter might be 1mm thick while having large focusing distance (especially when looking at chromatic aberrations..) - optics approach says you would not be able even get 1mm spot at such NA...
Why doesn't electrons widen the beam by repelling each other en-route?
Is there any good books on the subject explaining all these tricky stuff of focusing electron beam?
 A: The answer is a combination of two things:


*

*The "source size" (the first crossover size, in fact) is about 50 μm even for thermionic cathodes. They are able to do that by using hairpin cathodes and a negative electrode known as the Wehnhelt cylinder (see this image for a diagram of a thermionic electron gun).

*The electron probe can be made much smaller than the source by accepting a reduction in beam current. See page 30 and 31 of Reimer's Scanning Electron Microscopy for more details.


The reduction in current associated with a large demagnification is problematic, so field emission guns are preferred when small probe sizes are desired, because they have apparent source sizes in the nanometer range.
A: "Why doesn't electrons widen the beam by repelling each other en-route?"
This does happen and can cause unwanted distortions. The reason it's not an overwhelming problem is that the electrons, once they get going, are not all that close together. For example, if you have a 1nA beam current, and the electrons are traveling near the speed of light, there is on average 5cm vertical distance between one electron and the next. Also, once they're traveling very fast, it won't be very long until they to reach the sample, so even if they're slightly pushing each other, they can't get very far off track.
Everything gets worse when the beam current goes up, and also when the electron energy (hence speed) goes down. More specifically regarding electron energy: (1) When you lower the beam voltage, repulsion effects will get worse; (2) The beam path very close to the electron gun (before the electrons have accelerated much) is the part that needs to be designed most carefully to minimize repulsion.
I'm not an expert, this is my imperfect recollection from a few years ago. :-)
A: I am not a big expert in electron beams, but as long as nobody answers..
There is one thing which differs electron beams from the light. It is wavelength. It is few orders of magnitude smaller for electrons (~1Å) then for light (~5000Å for green light). Basically, there is one fundamental limitation for something to be focused. It is wavelength of this object.
However, it seems that your question is incorrect in a way that you have a wrong idea of an electron microscope. Where did you get these few tens of nm? Are you sure that you did not mix the resolution with beam focusing? Actually, for the best resolution beam should be defocused a bit. A good start is a wikipedia article.
A: If you want to focus the light of a 1 mm wide emitter to 0.5 mm wide spot, you can achieve that by making the lens to move towards the emitter. (at some fraction of speed of light)
In an electron gun we want to make the initial random speeds of the electrons very low, by making the electrode as cool as possible.
Then we could accelerate the magnetic lens towards the electrons, but it's easier to accelerate the electrons towards the magnetic lens. Anyway, the principle of relativity says we can achieve the same focusing effect either way.
So in this little analogy thermal electrons are analogous to thermal photons, and the speed of the lens relative to the light emitter is analogous to the speed of the magnetic lens relative to the electron emitter.
A quick course in information theory and thermodynamics of lenses and screens:
A lens-screen combination converts direction information into position information, an eye is an example of this.
A lens-screen combo converts order in the directions of particles into order of marks on the screen.
There are easy ways to remove disorder in directions of particles:
a: let the particles with more aberrant directions leave the group
b: accelerate all particles the same amount
Lenses are used to produce positionally ordered (narrow) beams, because it is easy to produce directionally ordered beams.
