- -- r. .-
Design of a 90’ Overmoded Waveguide Bend*
C. Nantista’ N. M. Krollr and E. M. Nelson
Stanford Linear Accelerator Center
Stanford University, Stanford, CA 94309 USA
A design for a 90” bend for the TEsr mode in over-
moded circular waveguide is presented. A pair of septa,
symmetrically placed perpendicular to the plane of the
bend, are adiabatically introduced into the waveguide be-
fore the bend and removed after it. Introduction of the
curvature excites five propagating modes in the curved sec-
tion. The finite element field solver YAP is used to calcu-
late the propagation constants of these modes in the bend,
and the guide diameter, septum depth, septum thickness, Figure 1. Outer geometry (a) and cross-section (b) of the
and bend radius are set so that the phase advances of all
bend. The cross-section’ dashed line is a symmetry plane.
five modes through the bend are equal modulo 2~. To a
good approximation these modes are expected to recom-
bine to form a pure mode at the end of the bend.
where C,,,, = Pm are the propagation constants and the
I. INTRODUCTION other C,,,, involve inner products of the transverse fields.
Some designs for the Next Linear Collider[l] (NLC) The power transfer between two modes in a curved
tr-ansmit power from the source (a klystron or the output section is limited by the difference in their propagation
of a pulse compressor) to the accelerator structure in the constants. The TEsl-TM11 degeneracy presents a prob-
TEei mode of overmoded circular waveguide in order to lem, so the degeneracy is split by introducing partial septa
. have small transmission loss. The waveguide run from the perpendicular to the bend plane as shown in Figure lb.
_ ._ -source to the accelerator includes some 90” bends. Ideally The modes can no longer be found analytically, but the Pm
these bends would be loss-less. can be computed using SUPERFISH[I].
Two algorithms and some results are presented for If pc ) d/2 then the coupling is weak and the TEer-
the design of one type of overmoded waveguide bend. A like mode amplitude varies little along the bend. A small
curved section of waveguide connects two straight sections amount of power will beat in and out of the nth coupled
as shown in Figure la. The curvature in the bend is con- mode in an arc length lb 1: 27r/]/?,, - /&I, where o indicates
stant so the waveguide follows a 90“ arc with radius of the TEsl-like mode. The interaction with each mode can-
curvature pe between the two straight sections. The cross- cels when the relative phase advance is a multiple of 27r.
section of the waveguide is uniform throughout the curved s
By adjusting the cross-section and p. the p’ are manipu-
section, but the cross-section is not simply a circle. The lated so that the three propagating modes coupled to first
cross-section and radius of curvature pe will be chosen so order all beat out at the end of the 90’ bend.
that the incoming wave propagates through the curved sec- This is the approach first taken. However, a compact
tion with negligible mode conversion. This is the principal bend which cannot rely on the above assumption is de-
form of loss considered here. Reflection and wall losses sired. As the coupling coefficients become comparable to
are only considered heuristically. The straight sections are the mode spacings, the beat lengths are altered, and modes
adiabatic tapers from and to circular waveguide. coupled to second order may be important. The coupling
S EQUATION coefficients C,, are required to verify parasitic mode sup-
pression at the end of the bend. Since the C,,,, are not
Curvature in overmoded waveguide causes coupling
easily obtained from the field solver, a different approach
between the straight guide modes. Such coupling is af-
forded by the generalized telegraphist’ equations, which
have been applied to curved circular guide. In terms of III. MODES IN CURVED GUIDE
the forward and .backward wave amplitudes, a$, these are
A curved guide can be treated as a portion of a cylin-
da* drically symmetric structure. For the 90” bend the struc-
2 = F:i C (C&a: + Cz,a,) , (1)
dz ture starts at 4 = 0 and ends at 4 = x/2. The fields
in the waveguide can be decomposed into modes with az-
*Work supported by U.S. Department of Energy imuthal dependence eim+. In the axisymmetric waveg-
contract DE-AC0376SF00515 and grants uide paradigm the waves propagate along 4 with propa-
DEFG03-92ER40759 and DEFG0392ER40695.
ovisitor from Department of Physics, UCLA, Los Angeles, CA gation constant tn. Compare this with the phase eia* for
90024. waves propagating along z with propagation constant p in
$ Also from Depsrtment of Physics, UCSD, La Jolla, CA 92093. straight waveguides. The curved guide does not close on
Presented at the Particle Accelerator Conference
(PAC 93), Washington, DC, May 17-m 1993
itself so there is n o requirement that m b e a n integer. a n d k = w/c is the drive frequency. Note that m is real for
T h e finite element field solver Y A P  is c a p a b l e of p r o p a g a t i n g m o d e s a n d imaginary for evanescent modes.
computing the frequencies of the m o d e s of axisymmetric T h e b o u n d a r y conditions E , = 0 at p = p C f w/2
structures for a n y real m. Non-integral m is allowed. Y A P yield a characteristic equation for the p r o p a g a t i o n con-
w a s u s e d to c o m p u t e dispersion d i a g r a m s for curved g u i d e stants m. Solutions w e r e obtained by numerically inte-
with various cross-sections. O n e such dispersion d i a g r a m is grating Bessel’sequations a n d using a shooting m e thod to
s h o w n in Figure 2. A dispersion d i a g r a m for curved g u i d e m a tch the b o u n d a r y conditions. This yielded numerical
looks similar to dispersion d i a g r a m s for straight guide. values for m 2 for both p r o p a g a t i n g ( m 2 > 0) a n d evanes-
However, the simple dispersion formula w2/c2 = kz + p 2 for cent ( m 2 < 0 ) modes. T h e field E , for e a c h m o d e w a s
a straight w a v e g u i d e containing n o m e d i a d o e s not apply obtained similarly.
to curved guide. This c a n b e s e e n best in figure 2, w h e r e T h e normalized generalized scattering m a trix Si w a s
the dispersion curves a r e not parallel lines. A p o w e r series c o m p u t e d for a n e x a m p l e with w/X = 1.36 a n d pc/X =
of the form 3.87, w h e r e X is the free s p a c e wavelength. T h e r e a r e two
p r o p a g a t i n g m o d e s in the guides. Using 1 4 m o d e s for the
w2 field e x p a n s i o n o n e a c h side of the interface, the c o m p u t e d
C2 scattering m a trix for the interface is
approximates the dispersion curves well. T h e cutoff k,” a n d
the coefficients oi d e p e n d o n p C a n d o n the cross-section R
of the guide. W h e n p C is large then cxi Cy 1 a n d the cut-
offs kz a r e approximately the s a m e b e t w e e n straight a n d
curved g u i d e with the s a m e cross-section. In the large p C
8.10-"/--4o & l o - & @ 1(3)
limit the two a p p r o a c h e s described in this p a p e r a r e equiv- w h e r e [a,~, usz, a,l, a,#’ is the incoming w a v e vector. T h e
alent. w a v e amplitudes as,, a n d a c n a r e for the m o d e s in the
straight a n d curved guides, respectively.
Notice that the reflection amplitude is less than 1 0 m 3 .
If o n e a s s u m e s the reflections a r e similar for b e n d s with dif-
ferent cross-sections but similar curvature, then reflection
at the straight-to-curved interface c a n b e neglected. T h e
reflected p o w e r will b e negligible as l o n g as r e s o n a n c e sa r e
avoided. T h e principal concern, then, is m o d e conversion.
V. AROUND THE BEND
T h e scattering m a trix S b for a b e n d over a n g l e C $ b
c a n b e easily c o m p u t e d given Si for the straight-to-curved
* .. "....'..." I interface a n d the p r o p a g a t i o n constants ml a n d m 2 for the
1 2 5 6
m 2 /f.$ (cm-:) two p r o p a g a t i n g m o d e s in the curved guide. T h e e x a m p l e
a b o v e h a s ml = 2 2 . 8 5 a n d m 2 = 16.18. T h e next m o d e is
Figure 2. Dispersion d i a g r a m of the curved g u i d e for the evanescent with m s = i11.38. T h e transmission coefficient
first design listed in T a b l e 1. T h e d a s h e d line is the drive for the (straight g u i d e ) fundamental m o d e for various b e n d
frequency 1 1 . 4 2 4GHz. T h e dotted line corresponds to the angles 4 6 w a s computed. A t & = 2?r/(ml - m 2 ) = 0 . 9 4 1
s p e e d of light a l o n g the center of the guide. the transmission is nearly perfect. A t this b e n d a n g l e the
two p r o p a g a t i n g w a v e s in the curved g u i d e arrive at the
IV .- S C A T T E R I N G A T T H E I N T E R F A C E output e n d of the b e n d with the s a m e relative p h a s e s they
T h e r e is potentially s o m e reflection at the interface b e - h a d at the input e n d of the b e n d . T h e p r o p a g a t i n g field
t w e e n the straight w a v e g u i d e a n d the curved waveguide. A at the output is the s a m e as at the input except for a n
generalized scattering m a trix Si for the p r o p a g a t i n g m o d e s overall phase, so w a v e s a r e faithfully transmitted through
in the straight a n d curved guides c a n b e constructed. the b e n d with n o m o d e conversion.
A s a n example, the scattering m a trix for the straight- T h e evanescent w a v e s at the interfaces h a v e d e c a y e d
to-curved interface in a n o v e r m o d e d rectangular H - p l a n e sufficiently in the curved g u i d e so that they c a n b e n e -
w a v e g u i d e b e n d w a s c o m p u t e d using a m o d e - m a t c h i n g glected in the transmission calculations for & = 0.941.
m e thod. O n l y TE,e m o d e s w e r e considered so the fields This e x a m p l e leads to the principal design criterion for
a r e uniform vertically. In the straight g u i d e p r o p a g a t i n g this type of o v e r m o d e d w a v e g u i d e b e n d : the p h a s e s e ”‘i + b
a l o n g y the m o d e s a r e E , cc sin(27rnzltu) w h e r e 0 5 x 5 w must b e identical for all m o d e s p r o p a g a t i n g in the curved
is the horizontal d o m a i n of the waveguide. In the curved guide. In addition, evanescent m o d e s should b e sufficiently
g u i d e the m o d e s involve Bessel functions. T h e y a r e E , cc a b o v e cutoff so that they d e c a y well over the length of the
A J m ( k p )+ B Y ,,,(b) w h e r ep c - w /2 L p 5 p c + w /2 b e n d , a n d thus c a n b e neglected.
- -- r. .-
90” Overmoded Waveguide Bends
d(cm) 1 (cm) w (cm) pc (cm) ml m2 m3 m4 m5 fc6 (GHz)
4.372 0.986 0.465 31.786 72.873 60.873 56.873 52.873 28.874 11.536
4.275 0.971 0.611 36.655 83.867 67.867 63.867 59.867 23.868 11.819
4.358 1.054 0.593 38.754 89.034 73.034 69.034 65.034 25.033 11.579
3.940 0.765 0.476 23.891 53.870 41.870 37.870 33.870 9.871 12.726
4.157 0.904 0.622 33.894 77.212 61.212 57.212 53.212 17.213 12.163
Mode 1 (“TErr”) Mode 2 (“TEsi”) Mode 3 (“TEsr”) Mode 4 (“TEer”) Mode 5 (“TMir”)
Figure 3. Electric field patterns for the five propagating modes of the first design in Table 1.
VI. 90” -BEND DESIGN and the decay amplitude over the length of the waveguide
is eimsa/2 = 5 x 10-8.
Designs for a 90” bend with a cross-section as shown in
Figure lb .w.e.r’ computed. The phases emin/z for the five
e VII. FURTHER WORK
lowest propagating modes excited by the incoming wave Further designs can be found, perhaps with smaller
can be fixed relative to each other by adjusting the four radii of curvature and shorter septa so that the bend will
parameters: d, pC, 1 and w. Propagating modes not excited have smaller wall losses and be easier to manufacture.
by the incoming wave (due to symmetry) are neglected. A variation of the YAP field solver will compute the
Dispersion diagrams were computed using YAP and the evanescent mode8 in curved guide. With these modes a
bend parameters were adjusted so that the phases were the mode-matching algorithm can be employed to calculate the
same. This corresponds to the propagation constants mi scattering matrix Si for the straight-to-curved guide inter-
differing from one another by multiples of 4. The cutoff face, and then verify that reflections are negligible and that
(m = 0) frequency of higher order modes were computed in the design criterion is appropriate.
order to discard designs with more than five propagating Calculation of the wall losses through the bend and
modes at 11.424GHz. Table 1 lists the parameters for mode-conversion losses (due to manufacturing errors) also
five solutions. It also lists the propagation constants for requires knowledge of Si in order to obtain the mode am-
the five lowest modes and the cutoff frequency fcs for the plitudes in the bend as well as the evanescent fields near
sixth lowest mode. the interface.
The cross-section in Figure 1 and the dispersion dia-
grams in Figure 2 correspond to the first design in Table 1. VIII. REFERENCES
The field patterns for the propagating modes are shown in [l] R.D. Ruth, “The Development of the Next Linear
Figure 3. At cutoff the field patterns for the modes in Collider at SLAC,” SLAC-PUB-5729 (1992).
curved guide are similar to the corresponding modes in s
 S. A. S ch elk unoff, “Generalized Telegraphist’ Equa-
straight guide, but for large m the second and third modes tions for Waveguides,” Bell System Technical Journal,
are mixed. This is evident in the field plots and in the 31, pp. 784-801, July, 1952.
dispersion diagram, where it appears that the second and  S. P. Morgan,“Theory of Curved Circular Waveguide
third curves are repelling each other. These modes arise, Containing an Inhomogeneous Dielectric,” B.S.T.J.,
with the introduction of the septa, from the TEz1 and TEsr 37, pp. 1209-1251, Sept., 1957.
modes of circular guide. The incoming wave is similar to  K. Halbach and R. F. Holsinger, Particle Accelerators
the fourth mode, which is a TEeI-like mode. 7, 213 (1976).
The cutoff frequency for the sixth mode of the first  E. M. Nelson, “A Finite Element Field Solver for
design appears close to cutoff. The estimated propagation Dipole Modes,” SLAC-PUB-5881, 1992 Linear Accel-
constant using the straight guide formula is me 2 i10.7 erator Conference Proceedings, pp. 814-816.