Stop the

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 havent 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 Hertzian 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 Maxwells 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 Maxwells 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 (Er and Ez are in phase)
• 2. the tangential E-field vanishes.

Will this lead to structures which are better
than what we have today?