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Stop the “bruits de couloirs”!

Stop the “bruits de couloirs”!. The quest for a “better” acc. structure: quasi-optical? The résumé in the beginning: Physics still holds. Accelerator text-books do not have to be rewritten. We haven’t found a “Super”-structure with lower surface fields than acc. gradients (yet).

lee-winters
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Stop the “bruits de couloirs”!

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  1. Stop the “bruits de couloirs”! The quest for a “better” acc. structure: quasi-optical? The résumé in the beginning: • Physics still holds. • Accelerator text-books do not have to be rewritten. • We haven’t found a “Super”-structure with lower surface fields than acc. gradients (yet). • but the numbers are quite OK for PETS (low r’/Q) In the following, I often normalize to the frequency (or wavelength). This allows easy scaling. For comparison, a standard CLIC structure has iris aperture a/l of 0.2, and a normalized period length p/l of 1/3. A typical value for r’/Q for the standard structure 25 kW/m, or 250 W/l. A typical ratio “surface field to acc. gradient” is 2.5.

  2. Bad “transit time factor” Parameters for this example: Shape: sphere, centered on axis. f: 27.361 GHz, l: 10.957 mm b: 10.207 mm, b/l: 0.932 a: 8 mm, a/l: 0.73 p: 14 mm, p/l: 1.278 vg: -34.55 % c Dj : 460 ° r’/Q: 533.4 W/m = 5.845 W/l gradient: 0.132 surface field: 0.291

  3. strange: field maximum under “iris”! • Parameters for this example: Shape: sphere, centered on axis. f: 30.868 GHz, l: 9.712 mm b: 10.207 mm, b/l: 1.051 a: 8 mm, a/l: 0.824 p: 14 mm, p/l: 1.441 vg: -3.09 % c Dj : 518 ° r’/Q: 403.3 W/m = 3.917 W/l gradient: 0.093 surface field: 0.312

  4. “Quasi-optical, high vg” • Parameters for this example: Shape: sphere, centered on axis. f: 43.652 GHz, l: 6.868 mm b: 10.207 mm, b/l: 1.486 a: 8 mm, a/l: 1.165 p: 14 mm, p/l: 2.038 vg: 67.78 % c Dj : 733 ° r’/Q: 27.8 W/m = 0.191 W/l gradient: 0.035 surface field: 0.63

  5. example with moderate aperture 0.65 Parameters for this example: Shape: ellipsoids, centered on axis. f: 29.981 GHz, l: 10 mm b: 9.41 mm, b/l: 0.941 a: 6.5 mm, a/l: 0.65 p: 6.666 mm, p/l: 0.6666 vg: -4.412 % c Dj : 240 ° r’/Q: 679.3 W/m = 6.793 W/l gradient: 0.159 surface field: 0.55 This is similar to the case which fascinated me.In opposite direction one gets (wrongly): 14.1 kW/m !

  6. “Inverse problem” Different approach: • Distribute Hertz’ian dipoles on the axis, spaced by the period, and properly phased, but in free space. • The vector potential has only a z-component and can be given analytically. • Calculate Er and Ez, and determine a possible slope for a metallic wall, i.e. rwall’(z) = - Ez/Er. • Choose a starting point and try to integrate to get a periodic solution for rwall(z). • This is not perfect, but see what I got. Parameters: Shape: special, from numerical integration. f: 30.358 GHz, l: 9.875 mm b: 10.64 mm, b/l: 1.077 a: 7.679 mm, a/l: 0.699 p: 13.333 mm, p/l: 1.35 vg: 5.556 % c Dj : 486 ° r’/Q: 447 W/m = 4.414 W/l gradient: 0.122 surface field: 0.44

  7. Yet another approach ... Start from Maxwell’s equations, for round, periodical solutions. Formulation with space harmonics: (normalized to f = c = e0 = m0= l = 1, so: k0 = w = 2 p)

  8. Space harmonic expansion Like in round waveguides (but not quite ...), we can separate r- and z-dependence. The field vector (Er, Ej, Ez) looks like where n runs over the space harmonics and p is the period. Note that only the space harmonic zero has net interaction with the beam. But... a “short” period p requires |kz| = 2p |n | / p to be larger than k0. With the separation condition k02 = kr2 + kz2 , this requires imaginary kr. This leads to modified Bessel functions ... Remark: When separating Maxwell’s equations for a round waveguide, you get something similar below cutoff, but there you require a real kr, so you you get an imaginary kz, whereas here you require a real kz, so you get an imaginary kr.

  9. Space harmonic expansion: radial dependence of fields This example is for p = 4/3 Note: space harmonic 0 has a constant axial and a linearily growing radial field! It is similar to a “radial cutoff” Space harmonics 1 and 2 have a real kr

  10. Can it be done? This describes exactly the fields which are required to accelerate particles in round periodic structures. It is in closed form, and has only parameters a(n) and the period p. The question is: playing with a few amplitudes a(n), keeping of course a(0) to a maximum can we synthesize a contour, i.e. find a line in r-z-plane where • 1.we have local linear polarization, i.e. field vector does not rotate (ErandEz are in phase) • 2. the tangential E-field vanishes. Will this lead to structures which are “better” than what we have today?

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