3-D computer simulations of EM field sin the APS vacuum chamber. Part 2: Time-domain analysis Page: 3 of 16
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The main results of these runs are shown in Figs. 2-5, and are explained below.
1. In run 1, the boundary conditions at the y = 0 plane allow for TMmi modes 1 but
eliminate T EmO modes in the narrow gap 2. The cutoff frequency of the TM waves,
i.e., the frequency of TM11 mode, of the gap is about 15 GHz and is much higher
than that of the beam chamber, which is about 4.6 GHz (see ). Therefore, we
do not see any significant field components in the gap, nor in the antechamber.
As shown in Figs. 2(a)-(c), the E fields in the beam chamber have a magnitude
of 1013 V/m, while that in the gap and in the antechamber are, respectively, five
and ten orders of magnitude lower. In other words, the TM modes do not couple
between the beam chamber and the antechamber, as expected.
Figs. 2(d) and (e) show the spectra of the E fields at probe 1 in the beam chamber.
It is seen that the spectrum of E, contains low-frequency componenets, which
are TE modes, whereas that of EZ starts at about 4.6 GHz, consistent with the
excitation of the first TM mode.
2. The results of run 2 are similar to that of run 1.
3. In run 3, the boundary conditions at the y = 0 plane do allow for TEmo modes
in the gap. The cutoff of TEno is low (about 0.33 GHz ). When the bunch
enters the beampipe, TM waves are generated and propagate. These TM waves
get scattered at the discontinuous corner between the beam holes and the beam
chamber. The scattering excites TE waves, which can then easily penetrate into
the gap as well as into the antechamber.
To convince oneself that this is a correct picture, let us compare the time delay
of the starting point, the first peak and the first big peak of the Ey field observed
at Probes 1, 3 and 5, with that calculated from the geometry. The calculations
are demonstrated in Fig. 3, in which the time needed for electromagnetic waves
propagating from one point to another is indicated in nanoseconds. The r.m.s.
bunch length, Qb, is 0.025 ns. Half of the total bunch length is 5 0-b, namely, 0.125
ns, which has to be taken into account. Let us now take Probe 3 as an example
to demonstrate our calculations. The distance between the entrance and Probe 3
is 0.47 ns. This is when the E-field starts to show up at Probe 3. As the bunch
center enters this geometry, which gives another time delay of 0.125 ns (5 oTb), we
see the first peak of Ey. When the bunch center reaches the discontinuous corner
between the exit beam hole and the beam chamber, the back-scattered waves will
enhance the fields at Probe 3 so that we will see a big peak as shown in Fig. 4(c).
The total time delay for this big peak is 0.8 ns (Fig. 3) plus 0.125 ns, i.e., 0.925 ns.
Table 1 gives a complete list of the results of this comparison. The good agreement
between the observation and the calculation supports the explanation given in the
'Here the notations of rectangular waveguides are adopted, namely, the subscripts of TM(TE)n,
stand for the number of half-waves in the z, y direction, respectively.
2The TEon modes are also allowed for in the gap. But we have no interest in them, because their
high cutoff frequency prevents them from penetrating into the gap.
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Chou, W. 3-D computer simulations of EM field sin the APS vacuum chamber. Part 2: Time-domain analysis, report, January 20, 1989; Illinois. (digital.library.unt.edu/ark:/67531/metadc791829/m1/3/: accessed January 17, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.