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                                                   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
                                    II. TELEGRAPHIST’
                                                    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-
                                                                                             was taken.
                             forded by the generalized telegraphist’ equations[2], which
                             have been applied to curved circular guide[3]. 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
     .Y        -
     --   t.
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    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 [5] 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
                                                                                          [ 0.190
                                                                                                                          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.

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                                                         Table 1
                                             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
                                                                    [2] 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            [3] 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            [4] 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          [5] 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.


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