Diffraction model of a step-out transition Page: 3 of 6
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DIFFRACTION MODEL OF A STEP-OUT TRANSITION *
A.W. Chao and F. Zimmermann, SLAC, Stanford, CA 94309, USAAbstract
The diffraction model of a cavity, suggested by Lawson
[1], Bane and Sands [2] is generalized to a step-out tran-
sition. Using this model, the high-frequency impedance is
calculated explicitly for the case that the transition step is
small compared with the beam-pipe radius. In the diffrac-
tion model for a small step-out transition, the total energy is
conserved, but, unlike the cavity case, the diffracted waves
in the geometric shadow and the pipe region, in general, do
not always carry equal energy. In the limit of small step
sizes, the impedance derived from the diffraction model
agrees with that found by Balakin, Novokhatsky [3] and
also Kheifets [4]. This impedance can be used to compute
the wake field of a round collimator whose half-aperture is
much larger than the bunch length, as existing in the SLC
final focus.
1 SHORT CAVITY (REVIEW)
The high-frequency impedance of a cylindrically symmet-
ric short cavity structure in an otherwise smooth vacuum
chamber pipe can be estimated by a diffraction model as
described in Refs. [1, 2]. It is helpful to first review this
model, before generalizing it to a step-out transition.
Let b be the radius of the smooth beam pipe, g the cav-
ity gap length, and d its depth. A schematic is shown in Fig.
1. As a beam current Jo = Jo exp(-iw(t-s/c)) enters the
3oe0m(t-sC)
d
Figure 1: Schematic view of a short cavity
cavity along the beam pipe axis, a diffracted wave is gener-
ated at the cavity edge r = b. This diffracted wave propa-
gates down the beam pipe while spreading out radially due
to diffraction. At a longitudinal distance s behind the en-
trance edge of the cavity, the radial spread of the diffracted
wave is about
1 As
Ay(s) 21r 2 (1)
*Work supported by the Department of Energy, contract DE-AC03-
76SF00515where A = 27rc/w. In the diffraction model of Refs. [1, 2,
5], the cavity gap length g is assumed to be short enough
that the wave has not spread out radially to reach the outer
wall of the cavity. Thus the outer cavity wall does not play
a role in determining the short range wake field, and the
quantity d does not enter the considerations.
In calculating the impedance of the above cavity struc-
ture, the wavelength A is assumed to be sufficiently short
that the diffracted wave populates only the radial region
close to r = b, and, in particular, neither penetrates much
into the depth of the cavity structure nor approaches the
pipe axis. In this case, one can approximate the cylindrical
geometry near the r = b region by a Cartesian geometry
and represent the incoming beam wave by a planar wave
with E, = -Bx = -2Jo/(cb). The monopole longitudi-
nal impedance is then found to be [2, 5, 6]Z01(w) = [1+sgn(w) i]Z 1 c
(2)
where Zo = 377 fl. The m $ 0 longitudinal and trans-
verse impedances Zmk and Zm are obtained by consider-
ing an mth moment current Jm = im exp(-iw(t - s/c)).
The corresponding planar wave incident upon the cavity
entrance is Ey = -Bx = - cb Jm cos mO, where 0 is
the azimuth at the cavity entrance. The final result is [2, 5]Z1(w) = [1 + sgn(w) i] Zc
X70/- b2m+l 01T(3)
The transverse impedance follows from the Panofsky-
Wenzel theorem, Z,(w) = cZ11 (w)/w. It has been shown
that exactly half of the diffracted wave energy is contained
in the geometric shadow region (outward diffracted), while
the other half is contained in the region propagating down
the pipe (inward diffracted) [2, 5].
2 STEP-OUT TRANSITION
Equations (2) and (3) apply at high frequencies and short
cavity lengths, when d ./Ag/2/(27r). The purpose of
this paper is to extend the 'conventional' diffraction model
just described to the case when the cavity gap length g is
long, and, thus, the above condition is violated. The cavity
structure in this case resembles a transition step in the vac-
uum chamber pipe. We further assume the step to be small,
i.e., b d, so that we can still approximate the cylindrical
geometry by a planar one.
Figure 2(a) displays the geometry near r = b. Shown
shaded is the region populated by the diffracted wave.2-
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Chao, A.W. & Zimmermann, F. Diffraction model of a step-out transition, article, June 1, 1996; California. (https://digital.library.unt.edu/ark:/67531/metadc694717/m1/3/: accessed April 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.