Nominally, it would be Gaussian-ish. Thing is, with this very high-current, short pulse, small-diameter beam there are all sorts of other intra-beam effects that can cause issues, which would show up if one could profile the beam as hoped. Also, a requirement for a FEL is very little velocity modulation. I mean, its already ultra-relativistic so all the electrons are going essentially the same speed. I should say, the beam energy that we are looking at is around 1 GeV (that is kinetic energy only).
Ah yeah, so this thing isn't exactly compact either; which...
the phase of the moon [really, I'm not making that up: https://www.aps.anl.gov/files/APS-sync/lsnotes/files/APS_1418251.pdf], etc ... ).
...There you have it.
Part of the motivation of my proposal was to do away with all of that space-consuming, expensive equipment that would be required (kicker magnet, new beamline with the expander magnets and imaging systems, etc ...). It would be really nice to profile the beam inside or between some short period undulators (say ~7-10mm), which have a gap between the jaws of ~3mm (yes, even smaller than my quoted 1cm distance).
Yeah, fairly compact, and very precisely aligned, shooting an immensely thin beam down many meters of beamline (and other structures).
And still, those structures (some ~mm tubes or pole pieces or etc.) are massive compared to the bunches, like 1000x bigger, give or take.
Huh, do the undulators have an exacerbating effect, tending to break up / disperse the beam? Is that part of the motivation?
Can they be made to undulate while still keeping the beam fairly well focused? (Maybe/probably not -- I think focus is a conservative thing, right? There's nowhere for positional energy to be sunk, except perhaps as synchrotron emission. It can work in the main beamline -- I think? -- by rotating the positional offset (transverse <--> axial), thus converting it to dispersion, which then exchanges energy with the accelerator fields, tightening the bunches (axially) without causing additional spreading; thus beam intensity goes up, rather than remaining constant or decreasing, whereas just magnets alone I think can only exchange it. Do I have that right?)
If I could measure two piece of information per position (magnetic field Bx and By, for instance) instead of just one (current or voltage), that would be twice as good, right? Other than that, no, I don't think magnetic fields would be any better. I just hadn't considered using the electric field.
But yes, at the moment, it seems like a collection of conventional beam position monitors is much more practically feasible. I hadn't thought of this, so thank you.
The thing is, if the electrodes are, say, a circular array of wires a few mm in diameter, but the beam cross section is some µm across, how can you really tell that the field at any given electrode, is due to the field of any given differential of the beam? Assuming superposition holds, of course (which, it ought to, anyway.)
Which is what I was talking about with fields, they're blurry at a distance, there's nothing to tell about something up close. You need electrodes on the scale of the beam itself, in which case you might not have a problem with dipping them into the beam directly (or it happens accidentally, anyway, because we're talking 10s of µm alignments).
Put another way, suppose the beam is misshapen with some elliptical section. If the distance from electrodes to beam are reasonably equal (everything's concentric), we might expect a small (potentially detectable?) quadruple moment aligned to the axis of that ellipse. Suppose the distribution is bimodal, two separate beams, one at each focus: could we even tell the difference? Or an elliptical shell, or anything else inbetween. And what if it's rotationally symmetrical, but the mean radius is 1um? 10? 100? Does the signal change at all?
We can potentially get information from EM fields, to the extent that it emits (again, at these frequencies) an optical image or whatever, and thus be diffraction limited instead. But, does it? As I recall, the thing with LINACs is, that's the whole point, if it's not accelerating it's not shedding radiation, it's propagating smoothly. I guess the implication is that, yes there's optical emission, but it's all virtual, and it can only be made real under certain conditions.
Also, how does that work, causally? Say the bunch moves past a metallic slot, so a current is induced (and some work done on its resistance): how does the bunch "know" it's lost some energy to that, since it's moving essentially at the speed of light? Oh, this is one of those advanced-retarded wave problems, isn't it; or equivalently, electrodynamics. Well, I shouldn't try thinking too hard about that, I think, but I'm not sure exactly where the "trust in the classical math, it's merely a pulsed current" line ends, and QM (or QED for that matter) picks up, and it seems like there might be something here. (Check the literature, someone must've thought of this before?)
So, I wonder if there could be something about -- imagine the bunch moving through space, emitting electric field lines like sonic shock cones. If they superimpose, and if we can assume the bunch length is much shorter than its width, then at a given, stationary, point detector, we have a time-domain representation, and can do something like 2D tomography to figure out the source's cross section. All we need is an ADC with sub-ps resolution!...
Also, for optical detectors, if we have coherent detectors (perhaps not outright impossible in such a compact, controlled environment?), we don't necessarily need to be diffraction limited, but can potentially do better (how much better, I don't know?). But this is an extremely fast (wideband) transient-field problem, can optics even be made to handle that? Let alone sample and downconvert it to a useful signal (something corresponding to beam cross section).
Oh right, that's what a coherent solution would be, using synchronous ADCs; and presumably it'd be fine to synchronize that in turn to the bunch rate, so that we can do something like equivalent time sampling of the beam profile.
There are optical techniques to essentially sample images with very short time apertures, so I guess that would be of some interest. I'm not sure offhand how to go from "current transformer" to "optical detector", and how to reconcile that with "unaccelerated beam doesn't emit". I think I've got some things inconsistent there, and I've just not done nearly enough particle physics (and in long enough time..) to know which...
Tim
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