Particle Accelerators, 1993, Vol. 41, pp. 173-201Reprints available directly from the publisherPhotocopying permitted by licence only

© 1993 Gordon & Breach Science Publishers, S.A.Printed in the United States of America

HIGHER ORDER MODE ANALYSIS OFTAPERED DISC-LOADED WAVEGUIDES

USING THE MODE MATCHING TECHNIQUE

URSULA VAN RIENEN

Technische Hochschule Darmstadt, Theorie elektromagnetischer Felder,Schloj3gartenstr.8, D 6100 Darmstadt, Germany

(Received 1 December 1992; infinalfonn 31 March 1993)

Several designs for linear colliders for e+e- -collisions in the TeV range are currently under discussion. Mostof them will use multi bunch operation. Therefore the study of higher order modes excited by previous bunchesin the train becomes even more important for the optimal design of the accelerator components. Many of thedesigns use long tapered disc-loaded waveguides for acceleration. Numerical reasons hinder-the mode analysis bydiscretization methods. In this paper the electromagnetic waves in tapered multi-cell structures with circular crosssection are calculated by a modal field matching method. The multi-cell structure is subdivided into subregionsof constant circular cross section. In each subregion the electromagnetic fields are expanded in orthogonal seriesover discrete modes. The field solutions are obtained by field matching at the transverse interfaces between thesubregions. The scattering matrix formulation is used to calculate the amplitudes of the field expansion. Someconvergence investigations are presented as well as comparisons with a grid-oriented numerical method and anequivalent circuit model. The field distribution of deflecting dipole-modes is analyzed for a 180-cell tapered disc-loaded S-band-structure of a 500 GeV Linear Collider Study.

KEY WORDS: Deflecting Modes, Aperiodic Waveguide Structures, Mode Matching

1 INTRODUCTION

In the future High Energy Physics will need very long linear colliders for e+e- -collisionswith multi-bunch operation to verify the laws of elementary particles with sufficiently highluminosity. Linear accelerators should be able to provide e+e- -collisions in the regionof 500 GeV to 1 TeVI since they do not suffer from the synchrotron radiation losses ofcircular accelerators. The luminosity of a machine, which is a measure for the rate ofcollisions, depends quadratically on the number of particles in a bunch and only linearlyon the number of bunches. Therefore, until recently, most investigations have been forsingle bunch operation. However a single bunch machine with a high number of particleshas problems caused by the strong wakefields. Consequently most designs currently underdiscussion for multi-bunch operation use parameters which are a compromise with regardto wakefields and other important effects.

A very important aspect in the design of accelerating components for these colliders istheir influence on the beam. One stringent requirement in the design of the accelerating

173

174 u. VAN RIENEN

structure is to avoid cumulative beam break up (BBU) which is caused by the (so-called)higher order modes. Mainly modes of the dipole passband are of greatest importance. TheBBU results from the excitation of transverse wake fields by previous bunches in a trainof bunches. This subject can be studied by the wake potential which is calculated fromthe transverse loss factors of the dipole fields. Besides the quantitative wake potential,qualitative knowledge about the field distribution of the modes inside the acceleratingstructure is also important in order to cure the BBU, e.g. by designing higher order modecouplers.

Simple circular waveguides cannot be used for particle acceleration because the phasevelocity ofall modes above the cut-off frequency is greater than the speed oflight. The well-known measure used to slow down the phase velocity to the velocity of the ultra-relativisticparticles is to load the waveguide periodically with irises.

Two important groups of iris-loaded waveguides are commonly distinguished: the constant-gradient and the constant-impedance type. A homogenous structure with constant iris- andcell-radii over the whole length is called a constant-impedance structure because the shunt-impedance remains constant over the structure. The electric field on axis of the acceleratingmode declines towards the end of a constant-impedance structure as power loss is taken intoaccount. A structure with constant electric field on axis can be obtained by appropriate ta-pering, i.e. iris- and cell-radius decrease over the length of the structure. Such a waveguideis called a constant-gradient structure.

Usually the electromagnetic properties of accelerator components are calculated by com-puter codes that discretize Maxwell's equations. For long tapered disc-loaded waveguideshowever, these methods would need the solution of extremely large algebraic eigenvalueproblems with many clustered eigenvalues. This is numerically difficult. This paper willpresent a different method to determine the electromagnetic fields, loss factors and relatedparameters of the higher order modes in the accelerating structures.

Until now it was thought that only the first dipole modes are the most severe ones inconstant-gradient structures and the dipole fields are expected to be trapped in only a smallpart of the structure. In order to damp the modes with the highest loss factors it is importantto calculate the fields in the complete structure.

While some ideas about the qualitative behaviour of the higher order modes in taperedmulticell structures could be proved for short models with strong tapering by using estab-lished grid-oriented computer codes like MAFIA3 or URMEL-T4 (e.g. Weiland et al. 2) it is necessary to use non-grid-oriented methods to calculate the fields in the completestructure with realistic dimensions. Equivalent circuits5 are one possible technique. Theseare fast methods but the influence of the chosen model on the results is not negligible sothat the results may be far away from the exact solution of Maxwell's equations. The modematching technique which has been chosen here is based on an exact formulation. In thechain matrix formulation this technique is often used for periodic structures (e.g. Esteban6

or Hahn7 ). Recently Heifets and Kheifets8 calculated monopole modes in an aperiodicstructure by mode matching technique. In this paper the mode matching technique will beused in the scattering matrix formulation for monopole and higher order modes in taperedstructures as it was first presented in van Rienen9 .

MODE ANALYSIS OF TAPERED WAVEGUIDES

2 MODAL FIELD MATCHING

175

In this section the theory ofmodal field matching will be described briefly. It is a well knownmethod in electrical engineering10,11; for more details see also Steinigke12 or Witting13 .Also in accelerator physics it has already been used14.

The Modal Field Matching Technique can be applied for structures which can be split intosubregions where analytic solutions of Maxwell's equations can be given as an expansionof an orthogonal series over discrete modes. The field solutions for the whole structure arethen obtained by field matching at the interfaces between the subregions. The placement ofthese interfaces is problem-dependent. The tapered multi-cell structures which are used foracceleration of electrons have in general to be simplified in order to apply a Mode MatchingTechnique. For the structure studied below all rounded comers were approximated by sharprectangular comers. Then the tapered structure can be subdivided by transverse interfacesinto subregions of constant circular cross section.

2.1 Mathematical Treatment of the Field Problem

Perfectly conducting metallic boundaries will be assumed. The structures are assumed to befilled with vacuum. The time-harmonica,b electromagnetic field fulfils Maxwell's equations

with the boundary conditions

rot E = -jwJjH,rot H= jWEE,div E = 0,div jj = 0

fi x E =0,ii· jj =0

(1)

(2)

(3)

(4)

(5)

(6)

on the lateral area of the waveguide.The general solution of this system of coupled differential equations (1),(2) is usually

derived for homogeneous waveguides by substitution of the vector potentials XCH) andACE) which are obtained from (3) and (4):

E(H) = rot XCH) ( H-waves )

jj(E) = rot ACE) ( E-waves ).

(7)

(8)

a For materials with constant E,J-L and "" Maxwell's equations lead to the wave equation. Separation of variableswith respect to time gives the Helmholtz equation for the function of spatial coordinates and the differential

equation of an oscillation with the solution e±jwt for the function of time. Here the convention e jwt is followed.

b Notice the list of symbols

176 U. VAN RIENEN

In general the vector Acan point in an arbitrary direction. In order to simplify the solutionof the system of coupled differential equations it is usual to choose one coordinate directionas the direction of the vector potential. In a circular waveguide the cylindrical coordinatesystem (p,

MODE ANALYSIS OF TAPERED WAVEGUIDES 177

2.2.1 Single Step in Cylindrical Waveguide The simplest example is to consider a stepin a cylindrical waveguide as shown in Fig. 1.

The coefficients Un(z) and In(z) in the Fourier-series, the so-called voltage and currentamplitudes, are connected to the amplitudes of the eigenwaves in positive and negativez-direction. In the regions I and II they are given asd

uI == rv· (aI e-jk;,n z + bI ejk;,n z )n Vhit· n nuII == fiJi· (bII e-jk;~7nz + all ejk;~7nz)

m Vhm· m mII == ry;. (aI e-jk;,n z _ bI ejk;,n z )n VLn· n n

If,[ = VYj/: (b~ e-jk~~",z - a~ ejk~~",z).

(15)

(16)

(17)

(18)

The transverse fields at the cross-sectional area between both volumes I and I I are givenby (13) and (14) with appropriate local coordinate z.

The continuity condition for the tangential fields E{,II (13) and ii{,!I (14) at thecommon cross-sectional area A I I leads to

00 00

~ UI e I == ~ UII ell~ n n ~ m m'n=1 m=1

00 00

~ II hI == ~ IIIhII~ n n ~ m m·n=1 m=1

(19)

(20)

Now expanding in the reciprocal direction (i. e. E{I is expanded in terms of the fieldeigenfunctions hI of the volume element I and ii{ in terms of the field eigenfunctions ellof the volume element I I; see Piefke10 ) assures that the power flow from both directionsonto the common cross-sectional area is the same. It also yields that only known boundaryvalues are used15. Exploiting the orthonormality of the eigenfunctions finally leads totwo matrix equations expressing the connection between the coefficients U and I by thecoupling matrices C and D:

VI == CVIIIII ==DII .

(21)

(22)

with the current and voltage amplitudes written in vector forme.The coupling matrices are only dependent on the geometry of the waveguide disconti-

nuity!. The formulation of the problem via the coupling matrix is especially well suited

d see list of symbols

e Note that these systems of equations are of infinite dimension.

! For a homogeneous waveguide the coupling matrices equal unity.

178 U. VAN RIENEN

A,- All '"

2b /AI"'f/ AIIa l all II" 1ft.. I ..E'flfEII

2a

b1 611" tic m..

HI 0 10 HII..

I I ,

:=0

FIGURE 1: Step in a cylindrical waveguide.

for periodic problems. To study the transmission or reflection behaviour of a waveguidewith discontinuities, it is more appropriate to use the scattering matrix formulation. Thescattering matrix is obtained from the coupling matrices by transformation (see appendix).

The scattering matrix connects the wave amplitudes a and b of the incident wave withthe reflected or transmitted waves respectively:

(23)

The scattering matrix depends on the geometry and the frequency. The transmission andreflection coefficients can be extracted from the scattering matrix.

MODE ANALYSIS OF TAPERED WAVEGUIDES 179

2.2.2 Disc-loaded Cylindrical Waveguide In a structure with multiple changes in diam-eter the scattering matrix is constructed for each subregion. These matrices can be concate-nated to obtain the scattering matrix for the total structure or parts of it.

The rule of combination is given for a structure of two sections. Let S(I) and s.(2) be twoscattering matrices to be combined, then the complete scattering matrix is given byg

8(1,2) - 8 8 8 - (S}~i + S}~iIRS}~is}Y,I S}~iIRS}~iI )- - -1 -2 - s}i1RTS}y'I S}iII + S}iIRTS}Y,IIS}~iI '

(24)

with R == (E-.sS2iS}~ II )-1 .The outgoing amplitudes hI and h II are calculated from thescattering matrix for the total structure. The scattering matrix of a structure with n sectionsis then given by

(25)

The inner amplitudes which are needed to determine the fields are calculated from thescattering matrices for the parts left and right of their location.

2.3 Field Calculation in Tapered Waveguides

The calculation splits up into two steps: First the resonant frequencies are determined witha mode matching code of Steinigke12 . As second step each of the resonant frequencies isstudied separately with the program ORTHO. This computer program has been developedin order to calculate the scattering matrices, inner wave amplitudes, electromagnetic fieldsand many interesting parameters for tapered waveguide structures. The exciting waveamplitudes at the left and right end of the structure as well as the frequency are input valuesfor the code.h , i

2.3.1 Standing Waves A computer program by K. Steinigke12 was used for the calcu-lation of the resonances. The resonant frequencies of a closed structure can be calculatedfrom the scattering matrix by determination of the zeroes of

(8 ± ES )det -1,1 - ~. -1,11

SII,I S.II,II ± E(26)

for perfect electric or magnetic boundary conditions according to the sign in front of E. Thedeterminant is a function of frequency since the transmission matrices 'sl,I and S.I1,1I and

9 Formally the individual chain matrices could be taken instead of the individual scattering matrices. The chainmatrix of the complete structure could then be obtained by simple multiplication but it could lead to numerical

instabilities which are avoided by using the scattering matrix formulation.

h In principle the program could also be used to investigate the transmission performance of an inhomogeneous

waveguide in some frequency range.

i Some modules of ORTHO are taken from Steinigke's code.

180 U. VAN RIENEN

the reflection matrices SI,II and SI1,1 depend on the frequency. This determinant vanishesif and only if the angular frequency w is a resonant eigenfrequencies Wr of the structure.Thus the program determines the zeroes of (26).

Now each resonant frequency is studied in the following way: The wave amplitudesat the ends of the structure (i.e. in the outmost irises) are taken as exciting amplitudes.Then ORTHO determines the scattering matrix and (inner) wave amplitudes and hence theelectromagnetic fields for the eigenfrequency Wr . Subsequently characteristic quantities asvoltage, power loss, loss factor etc. are determined.

In some cases, which are discussed in section 3.2, it is necessary to split up the completestructure at some iris inside: Let S(l) and S(2) be the two scattering matrices for the twoparts of the structure. In the separative iris both wave amplitudes, which are going in thesame direction, have to be equal for both parts. It follows that the resonant frequencies ofthe complete closed structure are found by determining the zeroes of

8(1) ± E 8(1) 0 0-1,1 - -1,118(1) (1) -E 0

det -11,1 SII,1I0 -E 8(2) 8(2)-1,1 -1,110 0 8(2) (2)-11,1 SII,1I ± E

(27)

The structure can be split in any arbitrary iris.

2.3.2 Travelling Waves To study travelling waves in tapered multi-cell structures thefirst step is to find the appropriate amplitudes at both ends, Le. the "exciting" wave vectors.Each travelling wave can be decomposed into two standing waves with magnetic or electricshorting planes at both ends.i This fact will now be used to find the correct amplitudes a I

and all at both ends of a tapered multi-cell structure. Let a(/L) and a(K) be the excitingamplitudesk of the two standing waves for a special mode. Then the amplitudes for thetravelling wave are calculated as

(28)

where the signs have to be chosen so that the wave travels in the right direction and thenormalizing factor 'TJ is chosen so that both standing waves have the same stored energy:'TJ == VU(/L) IU(K). Thus the exciting amplitudes a I and all from (23), which are input toORTHO, are determined according to (28). It is clear that a(/L) and a(K) have to belong tomodes with the same frequency.

j The superposition of standing waves gives, in general, also a portion of a reflected wave. But it is possible to

reach destructive interference of the undesired reflected portions.

k means either a I or a I I with that boundary condition

MODE ANALYSIS OF TAPERED WAVEGUIDES

3 IMPORTANT CHARACTERISTICS OF TAPERED WAVEGUIDES

181

The calculation ofdeflecting modes in tapered, disc-loaded waveguides is the main purposeof this paper. The travelling waves are treated as well since we wanted to exclude effectsintroduced by the geometric approximation of the waveguide. Thus the first step was tomodel this geometry in order to have the accelerating 21T'/3-mode. As second step thehigher order modes have been calculated for that geometric approximation of the waveguidestructure.

3.1 Travelling Waves

For periodic structures having a symmetry plane all resonant frequencies except the 0- and1T'-mode occur twice - once with perfect electric and once with perfect magnetic boundaryconditions. This is no longer the case in tapered waveguides. But frequencies can be tunedfor one particular mode by geometric variation of the iris- and cell-radii.

After tuning for a special mode, as e.g. the accelerating 21T'/3-mode in the S-bandstructure treated below, the exciting amplitudes for the travelling wave with this frequencycan be calculated by (28). Note that travelling waves in tapered waveguides in generaldo not show a constant phase advance from cell to cell. But Fourier analysis of thez-dependence of the fields indicates, in most cases, one dominant component in phasebehaviour.

3.2 Higher Order Modes

The main concern is to know the field distribution of these deflecting modes and to havesome quantitative measure for the interaction with the beam.

3.2.1 Field Distribution Another characteristic of tapered waveguides is the fact thatthe fields of higher order modes do not extend over the whole structure but are trapped inrelatively few cells. Modes with nearly vanishing fields at the left and right end cannot befound by determining the zeroes of (26) for the complete structure. The calculations forthose modes where the field is trapped in the inner cells, are done by splitting the structureinto two parts as described in 2.3.1, i.e. by determining the zeroes of (27). Usually it willbe necessary to choose several locations for the splitting in order to find all trapped modes.

3.2.2 Loss-factor The loss-factor16 is a measure of the interaction between the particlebeam and the accelerating structure: It measures the amount of energy that a particle cangain or lose respectively by the fields of a mode while passing through the structure. Theloss-factor is given as follows

IVl 2k(p) = 4U' (29)

182

k f (VfpC)

4

3

U. VAN RIENEN

homogeneous structure; dimensions= middle cell of tapered structuretapered 8truc~ure

2

1

4300 4500 4700 f / MHz

FIGURE 2: Loss-factor calculated by URMEL-T for the lowest dipole modes of a tapered structure with 18 cells

and for a periodic structure with 18 times the middle cell of the previous one.

V stands for the voltage seen by a particle which travels along the structure at radius p. Ustands for the stored energy of the mode. The loss-factorl is needed for the calculation ofthe wakepotential. (see appendix)

Another definition is often used which defines a loss-factor k', independent of the radiusp at which the voltage is taken:

k' == k(p) . _1_p2m

(30)

where m describes the azimuthal dependance of the mode.As already pointed out the tapered structure is matched for the accelerating 27r/3-mode.

All other modes have an aperiodic field distribution. This fact is of special interest for thehigher order modes. Since these parasitic modes extend only over part of the waveguidestructure their maximal loss-factor is ~smaller than in a corresponding periodic structure.However the sum of all loss-factors in one passband is about equal for both cases becauseof the law of energy conservation. Thus the aperiodic structure has more modes withsubstantial loss-factor while the periodic one has only one or two modes that interactstrongly with the beam. This general behaviour is displayed in Fig. 2. At SLAC, whichuses a 3m long S-Band structure of 84 cells, it was found17 that the most severe dipolemodes are "7r-like" modes which only extend over a few cells and are trapped in the firstpart of the structure. The question was whether this behaviour would also be found for

l The loss-factor k should not be confused with the kick-factor which has a different meaning.

MODE ANALYSIS OF TAPERED WAVEGUIDES 183

the S-Band structure of 180 cells with smaller cell-to-cell deviations in geometry than theSLAC structure.

This subject is very important in order to design features for damping these modes. Theso-called Higher Order Mode Couplers are outgoing waveguides whose cutoff-frequencyis above the one of the accelerating mode in order to couple out the parasitic dipole modes.If placed near the beginning of the structure they couple out modes with fields nearby. Thusmodes trapped near the middle of the structure are not damped by couplers near the firstpart.

4 VERIFICATION OF THE MODE MATCHING CODE

Even though the mode matching technique is used for many applications such as homoge-neous periodic structures (e.g. Esteban6, Hahn7), in the design of filters in order to optimizetransmission properties (e.g. Geib15 ) or for accelerator structures14,18 these formulationsare different from the one presented here for higher order mode analysis. In the followingthe results of ORTHO are compared with other methods ofcalculating the loss-factor. Otherverifications of the underlying formulation of mode matching can be found in Steinigke12

or Geib15 .Three structures will be considered in this section: two relatively short tapered waveg-

uides of 9 and 36 cells respectively and a 6m long S-Band-structure of 180 cells.The short waveguides were chosen for convergence investigations and for comparisons

with the grid-oriented code URMEL-T4 which uses a finite difference method.The loss parameters and the field amplitudes of the dipole modes in the 6m long S-Band-

structure are compared with the results obtained by an equivalent circuit model5 .

4.1 Convergence investigations

The mode matching technique yields by virtue of its methods the exact solutions ofMaxwell's equations. In order to use it for numerical field calculations the infinite Fourier-Bessel series (13) and (14) have to be truncated after a finite number of terms (i.e. modes).There are some well-known convergence criteria defined for periodic structures. Somestudies19 to check these criteria on tapered waveguides will now be presented.

4.1.1 Edge Condition The waveguide junction shown in Fig. 1 produces a sharp comerwhere the waveguide diameter changes. The electromagnetic fields show a singularityat this location. The solution of the boundary value problem is unique only if the so-called edge condition20 is included. The edge condition provides knowledge about theasymptotic decrease of modal amplitudes for some discontinuity problem in a waveguide.This available knowledge is lost for the modal coefficients of higher order when retainingonly a finite number of unknown modal coefficients in the field expansions. The workof Vassall021 reports on computations where this knowledge was taken into account.However this method was not used here since the gain was rather moderate comparedto the computational effort, even though it is interesting for calculations of the completeelectromagnetic field.

184 U. VAN RIENEN

frequency [ MHz]

2888 r---.......---------------~0,26'4

.-0,25I ,

2997 .. . -+- w-O,&

-.- w-O,75

0,2'4

0,16'-

0.1'4

O.06~

.....----~.g...-- ....-------4iP-------I]O..2991

-0,06"2888 .

-0,1"

10l'1210•e2887 ....... ......._--'-__.........._----. --a.__~

•number of modes in smallest iris

FIGURE 3: Resonant frequency of the accelerating 21r/3-Mode in a tapered 9-cell structure. The weightingfactor w for the mode ratio is the variable parameter.

4.1.2 Mode Ratio Consider again a simple step in a cylindrical waveguide as shown inFig. 1. The continuity at this step is evidently determined by the eigenvalues K~~(E,H)

and K~;{(E,H) because they characterize the transverse field. Suppose M eigenfunctionsare taken in section I and N eigenfunctions are taken in section II. The entries in thecoupling matrix measure the strength of the coupling between modes in both sections andthus the reciprocal influence on the wave-amplitudes. In order to take into account maximalinformation for a selected dimension (M or N given) the following geometrical criterionfor the mode ratio can be determined:

MN

a

b(31)

MODE ANALYSIS OF TAPERED WAVEGUIDES 185

0.12..a 0..

(8)0.01• .00

!

~~§lget~I

•{~

0.~§tJ,1G:.

§•iT-t-----.----...---..-----.-----.----------11 0..00 0.0- 0.01 0.12 0.. 0.20 o.,J,t L21 O.JZ~ z/m

~

E~'8~O

;;~.+------,---........--.......,r----......---..-----------

8-+~~Igq

~Iat~0 0....itj.G:0

-,-t------\-~__,___.,._,_--.------I LOG .... .... o.a ... 1.20 O.,J,t 0.21 e.JI~ 21m

~

E~'q~o

i-to.oo---.,....r----..,....r-----..Ta---..T.---e.JO,....---~....---- ...-----...(b)

FIGURE 4: Plot of the longitudinal electric field of the accelerating 21r/ 3-mode with and without filter for atapered 9-cell structure with M == 10 modes in the smallest iris and weighting factor w == O.75.

186 U. VAN RIENEN

This well-known criterion can be verified for the convergence of the accelerating mode ina tapered 9-cell structure as can be seen in Fig. 3. The weighting factor w is a factor for theratio, i.e. M/N == w . (a/b).

4.1.3 Gibbs' Phenomenon In the practical application of mode matching the continuitycondition is not exactly fulfilled since the Fourier expansion is truncated after a finitenumber of modes. This implies that the field strength will show discontinuities at thejunction of adjacent sections. From system theory it is well known that overshooting occursif a function with discontinuities is represented by a Fourier series. This overshootingcannot be eliminated by taking more modes into account.

The usual measure to cure this problem is to use a filter to weight either the eigenfunctionsor the coupling matrix. Here a cosine filter for the coupling matrix C == (cmn ) was used:

(m-l 1f) (n-l 1f)

c~n == Cmn . cos M _ 1 '"2 cos N - 1 '"2 . (32)

As expected the transverse and longitudinal fields became smoother than before as shownin Fig. 4. In consequence the loss-factor which results from integrating the longitudinalelectric field also converges better. Fig. 5 demonstrates this fact.

The effect of the filter seems not to be so strong for dipole as for monopole modes.

4.2 Performance of the computer code

One important aspect of every numerical method is the amount of computing time needed.The computing time needed by ORTHO is essentially determined by the matrix operations:the calculations of matrix elements, the matrix inversions but mainly the concatenations ofscattering matrices in order to calculate the inner wave amplitudes in each sectionm . Sincethe matrix dimensions depend quadratically on the number of modes which are includedin the expansion this quadratic dependence is reflected in the functional dependence ofcomputing time on number of modes19 . This is a real drawback of the method whichgives one reason for using it only in case that discretization methods or other reliablemethods cannot be applied. The calculations for the S-Band structure have been performedon the SIEMENS-NIXDORF-Computer S400/40 which is a high performance computerby Fujitsu with 5 GFLOPS. The calculations for one mode took about 30 minutes cpu. Forsmaller examples as 36 cells the calculation takes 3 minutes 27.5 seconds cpu on a SUNSPARC 1.

4.3 Comparisons with other methods

A tapered waveguide structure of 36 cells has been investigated by the equivalent circuitmodel5 and the finite difference code URMEL-T as well as by the mode matching codeORTHO. The iris- and cell-radii vary linearly between the end cells of the 500 GeV

m These are needed to determine the field values.

MODE ANALYSIS OF TAPERED WAVEGUIDES

k IV/pC) (FII.r • 0)

2,85

-3~

187

2,8 ---- .-0,76

-~

2,85 '--_--L-__....L.-_.-.L__-'--_---'__-'--__L--_--J -at.a 10 15 20 26 30 85 40 48

number of modes in the cells

klV/pC) (Filter • 1)

2,7 .

2,66 ~.~ .-tit

2,6 ...+.. .-0,6 -3~

---- .-0,75

-e- .-1,0 -4"-~ .-1,6

2.85 ~_---a__---L__--L.__....L.__........__..J-__..1.-_---J-at.

6 10 15 20 26 30 36 40 46

number of modes in the cells

FIGURE 5: Loss-factor of the accelerating 27f /3-mode with and without filter for a tapered 9-cell structure asfunction of the number N of modes in the cells. The convergence behaviour for different weighting factors w isdepicted. Filter=O means Filt~r off; Filter=1 means Filter on.

188

mode 2

Amplitude

u. VAN RIENEN

mode 21

Amplitude

cell number

mode 12Amplitude

cell number

mode 29Amplitude

cell number cell number

FIGURE 6: Some dipole mode patterns in a tapered 36-cell-structure calculated with ORTHO. The resonant

frequencies have been numbered in increasing order. The plots show the normalized (arbitrarily scaled) voltage

amplitudes. The abscissas give the cell numbers n=1, ... ,36.

S-Band structure, which is presented below. The main interest lays in the field patterns andloss-factors of the lowest dipole modes. Some typical field patterns are shown in Fig. 6.

4.3.1 Comparison with equivalent circuit model In the equivalent circuit model eachcell of the waveguide structure is represented by a resonant circuit. The complete structureis modelled by a chain of these resonant circuits. The modes of the circuit are computed by

MODE ANALYSIS OF TAPERED WAVEGUIDES 189

k / (VfpC)1.4

1

0.6

0.2

4100

.,~

., \ circuit model\'",, , ,

' ....,~- ' ....

mode matching

4300 4500 f / MHz

FIGURE 7: Loss-factor as a function of frequency for the 36 lowest dipole modes in a tapered 36-cell-structure.

The loss-factor has been calculated at the smallest iris radius (i.e. 1.17646 cm). The continuous line joins the 180

discrete dots which constitute the plot. Results of ORTHO and the equivalent circuit model are shown.

MAFIA. The coupling between the cells is performed by capacitive and inductive circuitelements22 • The currents are obtained by solution of an eigenvalue problem. Many othermodels of equivalent circuits are possible e.g. Bane23 •

Fig. 7 displays the loss-factor as a function of frequency, determined by ORTHO orthe equivalent circuit model, respectively. The differences which can be observed couldpossibly arise from higher band modes not taken into account in the equivalent circuitmodel. On the other hand the loss-factor is also a very sensitive parameter.Thus it is more important that both curves have the same order of magnitude, the sameproportion of dipole modes with high loss-factors and that the integral of both curves issimilar.

Fig. 8 and Fig. 9 show mode patterns for the S-band structure described in section 5. Theywere computed by the mode matching method or the'equivalent circuit model respectively.Again the differences are very likely caused by the small number of band modes in theequivalent circuit model. The loss-factor is displayed in Fig. 10 and Fig. 11.

The disadvantage ofthe equivalent circuit model is the very low number of dipole bandswhose effects are taken into account while it must be supposed that also higher band modeshave an important effectn . The model with few dipole bands assumes only a small couplingbetween the cells. Also the assumption that the structure differs only slightly from a periodicone from cell to cell restricts its application.

n Going to more and more bands reduces the speed of the method. Note that the mode matching is equivalent toa circuit model with infinitely many bands.

190 U. VAN RIENEN

~

Amplitudemode 11Amplitude

.1

~Amplitude

~Amplitude

~Amplitude

cell number

~

Amplitude

cell number

xnode...5.l.Amplitude

'- -ceii n~er -- '-

made..1lAmplitude

cell number

mode 81Amplitude

cell number

mode 91Amplitude

cell number

mode 101Amplitude

cell number

mode 111Amplitude

'",-, - -- - -- --cell number

cell number

cell number

--cell number

mode 141 mode 151Amplitude Amplitude

-wIIt~;j

mode 180Amplitude

cell number

mode 131Amplitude

mode 171Amplitude

1

.~-:=. --=-------,..---=--......,...,-~

cell number

cell number

mode 121Amplitude

cell number

mode 161Amplitude

cell number cell number cell number

FIGURE 8: Some mode patterns of dipole modes from the mode matching code. The resonant frequencies have

been numbered in increasing order. The plots show the normalized voltage amplitudes. The abscissas give the cell

numbers n=1, ... ,180.

MODE ANALYSIS OF TAPERED WAVEGUIDES 191

IIlQSklAmplitude

mode 11Amplitude

~Amplitude

mode 31Amplitude

.. -.. ,;- =' -;." .~._.-.:: ~.:. .:. -=--~- .....-=-~-~ -;---~._ ..:-._;::... -- '. ;;, :; -=--:--;;---~;---:;--.=. .. '. -:---~.... - ..... ---::;---cell number

~

Amplitude

cell number

~

Amplitude

cell number

~

Amplitude

cell number

............... - '.- ...... ~--::;--;:.---.-cell number

~Amplitude

cell number

mode 121Amplitude

cell number

mode 91Amplitude

cell number

mode 131Amplitude

cell number

mode 101Amplitude

cell number

mode 141Amplitude

cell number

mode IIIAmplitude

...

~: WNN/HM\\lNm~l~

cell number

mode 151Amplitude

cell number

mode 161Amplitude

cell number

cell number

mode 171Amplitude

cell number

cell number

mode 180Amplitude

.·l :. -=.':'- -:-._-;:;- -;;;- -;;--;:;-- ...cell number

cell number

FIGURE 9: Some mode patterns of dipole modes from the equivalent circuit model. The resonant frequencies

have been numbered in increasing order. The plots show the normalized voltage amplitudes. The abscissas give

the cell numbers n=1, ... ,180.

192

k / (V/pC)

2

1

4100

U. VAN RIENEN

4300 4500 f / MHz

FIGURE 10: Loss-factor k as a function of frequency calculated by the mode matching technique for the 180lowest dipole modes in a tapered 180 cell structure. The loss-factor has been calculated at the smallest iris radius

(i.e. 1.17646 cm). The continuous line joins the 180 discrete dots which constitute the plot. The "oscillation"

is caused by the simplified geometry with sections of 6 homogeneous cells, each. Thus the curve resembles

locally that one in Fig. 2 for a periodic structure. The curve for a structure tapered from cell to cell would be

correspondingly flatter.

k f (VfpC)

2

1

4100 4300 4500 f / MHz

FIGURE 11: Loss-factor k as a function of frequency calculated by the equivalent circuit model for the samestructure as in Fig. 10. The loss-factor has been calculated at the smallest iris radius (i.e. 1.17646 cm). The

continuous line joins the 180 discrete dots which constitute the plot. The "oscillation" is caused by the simplified

geometry with sections of 6 homogeneous cells, each. Thus the curve resembles locally that one in Fig. 2 for a

periodic structure. The curve for a structure tapered from cell to cell would be correspondingly flatter.

MODE ANALYSIS OF TAPERED WAVEGUIDES 193

4.3.2 Comparison withfinite difference method The finite difference code URMEL-Tdiscretizes Maxwell's equations on a triangular grid. The small deviations from cell to cellin the iris- and cell-radii cause no approximation problems for the triangular grid, whereasa rectangular grid would need a tremendous amount of grid points to achieve not only agood approximation of the geometry but also an algebraic problem which is numericallystable. Already a quite coarse discretization by URMEL-T can perfectly approximate thetapered geometry.

The functional behaviour of the loss-factor in dependance on the frequency was studiedfor meshes with about 3600, 22000 and 53000 mesh points, respectively. These calculationsindicate convergence towards the results obtained by ORTHO.Nevertheless the number of mesh points seems to be still far too small. The accuracy incovering the frequency range is strongly influenced by the fineness of discretization andtherefore gave one measure for the convergence. Still 53000 mesh points were alreadysufficient to observe the following: The mode patterns agree qualitatively in this respectthat the portion of cells with a special phase advance is about equal for most of the modes.Furthermore, approximately the same number of cells is filled with an electromagnetic fieldor field-free, respectively. Also, URMEL-T found several modes with high loss-factor andfield only in the rightmost cells of the structure as it was found for the S-Band structure byORTHO (compare Fig. 8. It is planned to study the 36-cell structure with more mesh points.

5 MODE ANALYSIS FOR S-BAND ACCELERATOR STRUCTURE OF 500 GEVLINEAR ACCELERATOR

Some more calculations will now be presented for the constant gradient structure of the 500GeV Linear Collider Study2.

The characteristics for the accelerating structure are given in Tab. 1.

accelerating frequencyaccelerating modewave lengthtotal lengthmean shunt impedanceattenuation factorQ-value of 21r/3-modegroup velocities

f= 2997 MHz21r/3-ModeA == O.lmL=6m (180 cells)r == 53.6MO/mT == 0.57 NeperQ ~ 13000vg == 4.1 ...1.3% . Co

TABLE 1: Table of characteristic parameters of the accelerating structure

Starting from the key features and the design of some test cells24 the geometry of the entirestructure was determined by some iterative process aiming for the right attenuation (fordetails see Witting13).

194 u. VAN RIENEN

Then the details of the geometry, which is taken for the calculations with the mode match-ing technique, had to be modulated: A somewhat simplified structure with 30 sections madefrom 6 equal cells and some coupling irises between these sections is used for the computersimulationo • In the modulation process a particular attribute of aperiodic structures had tobe considered carefully: The resonant frequency of a given mode is not automatically equalfor the magnetic and the electric boundary conditions, but the geometry has to be tuned toequality in frequency for the 27r/3-Mode. We have studied this modified geometry for theS-Band accelerator structure as given in Tab. 2 with f 21r/3 == 2997.364 MHz and attenu-ation T == 0.560 Neper.

,.....---

a i [cm] b i [cm] d [cm] g [cm] I [cm]

1.60616 ... 1.17646 4.06341 ... 3.92824 0.5000 2.8340 3.3340

TABLE 2: Table with geometry of studied S-Band structure.

This structure shows the linear variation of group velocity which is characteristic forconstant gradient structures. A phase shift of about 1200 was reached with some smalldeviations along the waveguide.

5.1 Resultsfor the Dipole Modes

Fig. 8 in the previous section shows the mode patterns for every tenth dipole mode in theS-band structure. Their characteristics are described here briefly:

• The first ones are 7r-like modes. The fields are trapped in the first 10% - 20% of thecells. They each have a high loss-factor as depicted in Fig. 10.p

• The next modes are trapped in inner cells. Their loss-factors are still very highbecause the portion of 5-10 cells with 7r-like field pattern dominates.

• Then follow some modes which are mainly trapped near the end of the structureand have a very small loss-factor.

• The last modes are either trapped near the end of the structure or they are mixedmodes, i.e. they have a portion of a mode of the first passband at the end of thestructure and another portion of a mode of the second passband in the first part ofthe structure.

This result was unexpected and influences the design plans for damping or tuning strategiesin order to treat the deflecting higher order modes.

°This technique is well known from existing 5-6 m long structures in injector linacs.p The loss-factors have been calculated at the smallest iris-radius.

6 SUMMARY

MODE ANALYSIS OF TAPERED WAVEGUIDES 195

The application of the Mode Matching Technique to calculate the fields and quality factors(especially the loss-factor) of monopole or dipole modes in a long (e.g. 180 cells) taperedmulticell structure ofcylindrically symmetric, pillbox-like cross section has been presented.The exciting (i.e. outer) amplitudes for travelling waves, as e.g. the accelerating mode insome linear accelerator, are calculated from the corresponding standing wave problem. Formonopole and dipole modes some results have been presented.

Some comparisons were made for a tapered 36-cell structure. The comparisons with thefinite difference code URMEL-T indicate that a good agreement with the calculations donewith the mode matching code will probably be found with finer discretization. Comparisonswere made between an equivalent circuit model and the Mode Matching Technique forthe 36-cell structure and for the slightly modified geometry of a constant-gradient S-Bandstructure with 180 cells. Both comparisons showed the same peculiarities. The S-Bandstructure is designed for a 500 GeV Linear Accelerator. The interesting but unexpectedresult was that 2/3 of the first 180 dipole modes have a high loss-factor. Consequently, mostof these higher order modes are "severe" modes. As expected, most of the dipole modes aretrapped in a few percent of the complete structure. However, not only those modes, whichare trapped in the first part of the structure, are among the severe modes but also modeswhich are trapped in the middle part. These results contradict the usual ideas about thebehaviour of higher order modes in tapered waveguides. They will probably influence thedesign plans for damping. strategies for the S-Band structure. The results of the equivalentcircuit model differ slightly from those results obtained by the mode matching code. Thiscan be explained by the fact that in the equivalent circuit model higher band modes are nottaken into account and that only a small coupling between the cells is .assumed.

One important conclusion of the work presented here is that it is very important to dohigher order mode analysis for the complete structures rather than to study only very shortmodels because the electromagnetic behaviour of both can differ considerably.

ACKNOWLEDGEMENTS

This work was performed in a study group for a 500 GeV Linear Accelerator2 . I wish tothank T. Weiland for the suggestion of this work, him, K. Steinigke and my colleagues formany helpful discussions, K. Steinigke for his resonator code using the mode matchingtechnique, M. Witting for many of the calculations presented here, J. Nahr for some con-vergence investigations and S.G. Wipf for careful reading ofthe manuscript.

REFERENCES

1. Fourth International Workshop on Linear Colliders, Garmisch-Partenkirchen, Germany, August 19922. T. Weiland et al., Status Report of a 500 GeV S-Band Linear Collider Study, Proceedings of the

Workshop on Physics and Experiments with Linear Colliders, Saarise1ka, Finnland, September 19913. T. Weiland, On the Numerical Solution ofMaxwell's Equations andApplications in Accf!leratorPhysics,

Particle Accelerators 17 (1985) 227.

U. VAN RIENEN

U. van Rienen, T. Weiland, Triangular Discretization Method for the Evaluation ofRF-Fields in Wave-guides and Cylindrically Symmetric Cavities, Particle Accelerators 20 (1986/87) 239.H.-G. Beyer, M. Drevlak, N. Holtkamp, U. van Rienen, V. Tsakanov, R. Wanzenberg, T. Weiland,Z. Min, Attenuation of transverse modes by variable cell geometries in travelling wave tubes, XVthInternational Conference on High Energy Accelerators, Hamburg, Germany, July 1992, p. 876-878J. Esteban, J. M. Rebollar, Characterization of Corrugated Waveguides by Modal Analysis, IEEETrans.on Microwave Theory and Techniques, Vol. 39, No.6, June 1991, p. 937-943H. Hahn, On the Analysis ofPeriodic Waveguide Discontinuities by Modal Field Matching, ABU Band32, Heft 2, 1978, p. 81-85S.A. Heifets, S.A. Kheifets, Longitudinal Electromagnetic Fields in an Aperiodic Structure, SLAC-PUB-5907, September 1992U. van Rienen, Mode Matching Technique for Calculating Multi-Cell Cavities, Third InternationalWorkshop on Linear Colliders, Protvino, USSR, September 1991, Vol.3, Issue 2, p. 352-365G. Piefke, Feldtheorie III, (english: Field Theory II/), BI-HochschultaschenbUcher, Bd.782, Bibli-ographisches Institut AG, Mannheim, 1977T. Itoh (ed.), Numerical Techniquesfor Microwave andMillimeter-Wave Passive Structures, John Wiley& Sons 1989, p. 592-621K. Steinigke, Wellenausbreitung in koaxial und exzentrisch geschichteten Rundhohlleiterstrukturen,(english: Propagation of Waves in Concentric and Eccentric Multilayered Circular Waveguide Struc-tures), Fortschritt-Berichte R. 21, Nr. 108, VDI Verlag, DUsseldorf, 1992This PhD-thesis studies the propagation and scattering of electromagnetic waves in waveguides whichare loaded with lossy dielectric insertions. Arbitrary many layers as well as arbitrary many linediscontinuities in the direction of propagation are possible. The solution of the electromagnetic fieldproblem at the line discontinuities is obtained by the mode matching technique.

13. M. Witting, Untersuchung elektromagnetischer Eigenschaften von Wanderwellenrtihren mit der Metho-de der Orthogonalentwicklung, (english: Investigations on the Electromagnetic Properties ofTravellingWave Tubes by virtue of the Mode Matching Technique), Diplomarbeit am FB 18, FG: TEMF der THDarmstadt, Mai 1992This Diplom-thesis describes the application of the mode matching technique (here the programORTHO) to the S-Band tube, which is described in this paper. The tube has been modelled for thefield calculation in order to reproduce the accelerating 21r/ 3-mode. The technique of splitting thecomplete structure in two arbitrary parts has been described. Several dipole modes are calculated.

14. K. Bane, B. Zotter Transverse Modes in Periodic Cylindrical Waveguides, XIthe International Confer-ence on High Energy Accelerators, Geneva, Switzerland, July 1980, p. 581-585

15. B. Geib, Rechnergestiitzter Breitbandentwuif in Rechteckhohlleitertechnik, (english:Computer-Aided Wide-Band Design in Rectangular Waveguide Technique), PhD-Thesis, TechnischeHochschule Darmstadt, Darmstadt, 1992This PhD-thesis treats the design of wide-band microwave filters which are built from coupled cavitiesin rectangular waveguide techniques. Basic network elements are rectangular waveguide discontinuitiesand branches with variable height and width. Step-like profiles approximate continuous changes of thecontour. The electromagnetic field problem is solved by the mode matching technique.

16. P.B. Wilson, Wake Fields and Wake Field Acceleration, AlP Vol.127 (1983), p. 875-92817. R.B. Neal (General editor), The Stanford Two-Mile Accelerator, Stanford University, W.A. Benjamin,

Inc.,New York, 196818. E. Keil et al., Beam-Cavity Interaction in Electron Storage Rings, NIM 127, 1975, p. 475-48519. J. Nahr, Konvergenzuntersuchungen zur Orthogonalentwicklung in nichtperiodischen Wanderwellen-

rtihren, (english: Convergence Investigationsfor Mode Matching in Aperiodic Travelling Wave Tubes),Diplomarbeit am FB 18, FG: TEMF der TH Darmstadt, August 1992This Diplom-thesis reports about the convergence behaviour of the mode matching code ORTHO. Somerelatively short (e.g. 9 cells) tapered structures have been studied.

20. J. Meixner, Die Kantenbedingung in der Theorie der Beugung elektromagnetischer Wellen an vollkom-men leitendem ebenem Schirm, (english: The Edge Condition in the Theory of the Scattering ofElec-tromagnetic Waves at Peifectly Conducting Plane Screen, Ann. der Physik, Folge 6, Bd. 6 (1949), p.2-9

196

4.

5.

6.

7.

8.

9.

10.

11.

12.

MODE ANALYSIS OF TAPERED WAVEGUIDES 197

21. C. Vassallo, On a Direct Use ofEdge Condition in Modal Analysis, IEEE Trans. MTT, Vo1.24, April1976, p. 208-212

22. T. Weiland, B. Zotter, Wake Potentials of a Relativistic Current in a Cavity, Particle Accelerators 11(1981) 143.

23. K.F. Bane, R.L. Gluckstern, The Transverse Wakefield of a Detuned X-Band Accelerator Structure,internal report SLAC-PUB-5783, March 1992

24. H.-G. Beyer, M. Drevlak, N. Holtkamp, U. van Rienen, V. Tsakanov, R. Wanzenberg, T. Weiland, Z.Min, Automatic Computer-Aided Optimization of Cavities for the Design ofAccelerating Structures,XVth International Conference on High Energy Accelerators, Hamburg, Germany, July 1992, p. 939-941

APPENDIX

Field expansion in circular waveguide structures

Some more details about the mathematical background of the field expansion and modematching are given in this appendix.

Orthonormality of the field eigenfunctions en (p,

(37)

(38)

(39)

198 U. VAN RIENEN

Power Loss The Ohmic losses by wall currents in a dissipative material are given asfollows:

P == f J J* dV == JNtH; dA,~ 8~

where 8 stands for the skin depth, ~ for the conductivity and A is the surface of the material.The factor 1/8~ == Jw{t/2~ can be put in front of the integral.

Phase Constant, Phase Velocity The phase constant (3 is defined from the wavelength ofa mode by

f3 = 211"Az

That mode in a periodic structure with (3 . l == 21r/ 3, where l is the period length of thestructure (i.e. cell length plus iris thickness) is called 21r/3-mode. The phase velocity of atravelling wave is given by

WVph == ~.

A wave can be used for electron acceleration if the phase velocity is about equal to thevelocity of light. In a constant-gradient waveguide structure the phase velocity is constantalong the structure only for the accelerating mode.

Attenuation The attenuation factor T is defined by

-27 P Le == Po .

where Po is the input power, PL the output power.

(40)

Voltage For a particle which travels along the structure at a radius, p 2:: 0, with speed oflight, the voltage is defined by

(41)

Q-value An important parameter in the design of accelerating structures is the Q-value.If W is the time-averaged energy which is stored in the field and P the power loss, theQ-value is defined by

Q == _ wWp. (42)

While the shunt-impedance r == V 2 / P L (L=length of the structure) and the Q-valueremain constant for every cell of a constant-impedance structure the behaviour differs for

MODE ANALYSIS OF TAPERED WAVEGUIDES 199

the accelerating mode of a constant-gradient structure: the shunt-impedance grows slightlyand the Q-value shows a minor decrease along the structure.

Wakepotential Consider an exciting particle passing the structure on the z-axis withvelocity c and a test particle moving with the same velocity in a distance s on the samepath. The exciting particle generates Wake Fields which produce a Lorentz-force acting onthe test particle. The integral effect of this force is described by the Wakepotential:

"W1I(s) == 2I:k~cos(wnS/C), for s > o.n

(43)

in terms of the loss-factors k~. The transverse and longitudinal Wakepotential are relatedby the Panofsky-Wenzel theorem

a"W1- ( s) n W ( )as == v 1- II s .

Group velocity The group velocity of a mode is given by

(44)

(45)dw

V g == d(3

for a periodic structure, as a constant impedance structure e.g.. Group velocity and energyvelocity are equal for a lossless structure. The group velocity decreases approximatelylinearly in constant-gradient structures in order to compensate the field variation due topower dissipation. The fields of a travelling wave can be calculated including the powerloss effect by the damping term e-w /(2v g Q)z == e-P /(2v g W)zq.

q This feature is also implemented in the code ORTHO which is presented here.

200

LIST OF SYMBOLS

U. VAN RIENEN

All symbols are listed except symbols with clear conventional meaning as e.g. the normalvector n.

Symbol

AaabbCCCoCmn

c~nDdEEe

rS.T

Description

vector potentialradius of the iris holewave quantityradius of the cellwave quantitynormalizing constantcoupling matrix of voltage amplitudesvelocity of light in vacuum (1/VEOJ.Lo)coupling coefficientfiltered coupling coefficientcoupling matrix of current amplitudesthickness of irisvector of the electric field strengthunit matrixfield eigenfunction of the electric fieldunity vector in z-directionfrequencycell lengthvector of the magnetic field strengthfield eigenfunction of the magnetic fieldlocal current amplitude or also index for part of volume (s. b.)Bessel function of order J.L

An-th non-vanishing zero of the Bessel function of J.L-th ordern-th non-vanishing zero of the derivative of the Bessel function of J.L-th ordersectional eigenvalueloss parameterwavenumberpropagation constant of a waveguide modetube lengthperiodic length of a tubenumber of waveguide modes taken into account for the series expansionnumber of waveguide modes taken into account for the series expansionpowerQ-factorfunction for longitudinal dependance of the vector potentialshunt impedance per unit lengthscattering matrixsectional eigenfunction of the vector potential

MODE ANALYSIS OF TAPERED WAVEGUIDES 201

U local voltage amplitude or stored energyV accelerating voltageVE energy velocityvg group velocityVph phase velocityW stored energy per lengthW Wakepotentialy characteristic wave admittanceZ characteristic wave impedancea damping constant{3 phase constant of a travelling waveEO permittivity of the vacuum (8.854 . 10-12 AsNm)

'TJ normalizing constant

"" conductivityAz wavelength in direction of propagationIl index for azimuthal dependance

Ilo permeability of the vacuum (1.256 . 10-6 VrnlAs)1r circular number(p, 4', z) cylindrical coordinate systemT damping factorD.¢ phase advance per cellw circular frequency

indices: X arbitrary number, vector, ...X-I inverse matrixX(H) X(E) H-wave respectively E-wave,XI,XII I, I I to index elements of volumeXI, XII indices of elements of volume may be noted soXm,Xn m, n to index waveguide modes or travelling wavesX r resonanceX

T transposed matrixX t component perpendicular to direction of propagationX z component in direction of propagationX(JL) , X(~) index for perfect magnetic or electric terminationX~¢ index of travelling waves (e. g. D.¢ = 21r / 3)