Accelerating cavities exchange HOM power through interconnecting beam pipes in case of signal frequencies above the cut-off of their propagating waveguide modes. This may lead either to improved HOM damping or - in the case most severe - to unwanted phase coherence of fields to the beam. Therefore the knowledge of the scattering properties of a cavity as a line element is needed to analyse all kinds of RF cavity-cavity interaction. Since there is a lack of measurement tools capable to provide a multidimensional scattering matrix at a given frequency point, we have been developing a method for this purpose. It uses a set of 2-port S-parameters of the device under test, embedded in a number of geometrically different RF environments. The application of the method is demonstrated with copper models of TESLA cavities

Glock, Hans-Walter

Hülsmann, Peter

Klein, Horst

Marhauser, F.

English

Hochschulschriftenserver - Universität Frankfurt am Main

ANALYSIS OF HOM INTERACTION BETWEEN CAVITIES BYMULTIMODAL S-PARAMETER MEASUREMENTSH.-W. Glock, F. Marhauser, P. Hülsmann, M. Kurz, H. KleinInstitut für Angewandte Physik der Johann Wolfgang Goethe-Universität Frankfurt am MainRobert Mayer-Straße 2-4, D-60054 Frankfurt/M., GermanyAbstractAccelerating cavities exchange HOM power throughinterconnecting beam pipes in case of signal frequenciesabove the cut-off of their propagating waveguide modes.This may lead either to improved HOM damping or - inthe case most severe - to unwanted phase coherence offields to the beam. Therefore the knowledge of thescattering properties of a cavity as a line element is neededto analyse all kinds of RF cavity-cavity interaction. Sincethere is a lack of measurement tools capable to provide amultidimensional scattering matrix at a given frequencypoint, we have been developing a method for thispurpose. It uses a set of 2-port S-parameters of the deviceunder test, embedded in a number of geometricallydifferent RF environments. The application of the methodis demonstrated with copper models of TESLA cavities.1 INTRODUCTIONUsually the spectra of beam-excited fields contain signi-ficant parts above the RF cut-off of several waveguidemodes of the beam pipe. Therefore one has to be con-cerned about the effects caused by the RF-interaction ofthe beam line elements: it may for example increase thedamping of resonant cavity modes by stronger coupling toabsorbing elements, it allows for resonant fields localisedbetween different beam pipe insertions or it may introducesignal paths delayed by multiple reflection to - in theworst case - particle velocity. Either of these situations iscovered within a description in terms of transmission andreflection properties of the beam line´s elements withrespect to their interconnecting planes, i.e. their scattering(S-) parameters. Since the interaction is not restricted tothe fundamental mode, but includes a - frequency depen-dent - number of waveguide modes, the multidimensionalmatrix of the S-parameters (see sect. 2) is necessary andsufficient to characterise a line element in any environ-ment. Thus we have been developing an experimentalmethod, allowing for multidimensional S-matrix measure-ment in a waveguide setup. Since its first presentations[1,2] we adopted an improved evaluation scheme whichminimizes the overall error. It is demonstrated in the caseof two modes for a device measurement and three modesfor a calibration of an adaptor at single frequency points.2 THEORYThe S-matrix of an adaptor (A, B in Fig. 1) with onecoaxial line (index 0) and n modes propagating in thewaveguide port may be written as: A =A00 A01 A0nA01A0nA11 A1nA1n Ann=A00 ATA A(1)Herein the scalar A00 describes the reflection at the coaxialport, A  the coupling from the coaxial line to each wave-guide mode and A the reflection at the waveguide flange,that may couple every mode to each other. The matrix issymmetric due to the reciprocity of the device. Like in thesingle mode case, the problem of determining the proper-ties of a device splits into the calibration step - i.e. deter-mination of the adaptors, which connect the device to anideal network analyzer - and the measurement once theadaptors are known. Considering the number of unknowns(10 for two modes at two waveguide ports) it becomesclear that a single measurement with two completelyknown adaptors, which gives three numbers (two reflec-tion, one transmission quantity), not provides sufficientinformation. Thus one has to use different pairs of knownadaptors for a number of subsequent device measurements.To keep the calibration effort as small as possible we taketwo adaptors and then combine them with various delayline lengths (Fig. 1). In the same manner we use a shortand different delay line lengths (which are the only reliablebroadband waveguide standards of easy making) tocalibrate the two adaptors.A L1a0b0(-1)a1…n b1…n          B L2h0g0 (-1)g1…n h1…nAL1 CBL2a1…n b1…n g1…n h1…nc1…n d1…n e1…n f1…na0b0h0g0Figure 1: Schematic drawing of setups used for adaptorcalibration with delayed shorts and for measurement.Small letters denote the signals at all connection planes,index 0 corresponds to the coaxial line.If we consider a setup with two adaptors A and B, a testdevice C and two connecting waveguides with lengths L1and L2 (Fig. 1), we are able to write down all signals,related by S-matrices:521 0-7803-4376-X/98/$10.00 Ó 1998 IEEE  b 0b=A00 ATA Aa0a ,   g0g =B00 BTB Bh 0h  ad= 0E (L1)E(L1)0bc,   eh= 0E (L2)E(L2)0fg  cf=C11 C12C12T C22de (2 a-e)All the submatrices are (n x n)-dimensional. For thewaveguide of length L and the phase constants gj holds:  E(L) =e– g1 L 000 000 e – gn L(3)Using an additional abbreviation  H11 H12H12T H22=E(L1)00E (L2)C11 C12C12T C22E(L1)00E (L2)and with respect to the fact, that the complete setup isjust a coaxial line - two port with a (2 x 2)-S-matrix   b 0g0= G1 TT G2a0h 0 (4)one finds after elimination of all signal quantities ([1]):   G1 TT G2=A00 00B 00++ AT 00BTH11 H12H12T H221 –A 00BH11 H12H12T H22(–1 )A 00B(5)We shall refer to (5) as the "complete model". It can berewritten using  1 –M(–1 )=1 +M+M2 +M3 + ... (6)(we skip the discussion of the mathematical conditions)   G1 TT G2=A00 00B 00++ AT 00BTH11 H12H12T H22A 00BH11 H12H12T H22jSj=0¥A 00B(7)as a geometric matrix series. This expansion is useful aswell as an approach for the numerical solution of (5) witha set of measurement data as for its physical interpre-tation. We are going to use an approximation for the righthand side of (7), that contains only the terms of order j=0and j=1. This will be called "reduced model". To simplifydiscussion, we restrict ourselves to the calibration step,which is a special case of (5) (set all elements of C to 0except the upper left block which is the negative identity).Then the complete model is:   G1(L1)=A00 –AT E2(L1)1 +AE2(L1)(–1 )A (8)and its reduced version reads like:   G1(L1) » A00 –ATE2(L1)A+ATE2(L1)AE2(L1)A (9)Evaluating this for two modes (10) shows that each termdescribes a possible signal path from initial incidence tofinal detection. Thus the arithmetic derivation of (5) ((7)resp.) is just a summation over all paths, written verycompact.   G1(L1)=A00 –A 012 e–2ig1L1 +A 022 e–2ig2L1 ++A 012 A11 e–4ig1L1 + A022 A22 e–4ig2L1 ++2A01 A02 A12 e–2i g1 + g2 L1 – (10)In order to find the S-parameters of C (A, B resp. duringcalibration), a set of equations (5) ((8) resp.) with mea-sured data inserted has to be solved. This is done by aniteration scheme, which significantly improves the resultscompared with the approximative approach [1,2]. Thelatter one remains as a first step of the method, used tofind initial values in the following way: Neglecting allhigher orders of internal reflection but the first,   G1 L1,L2 (and similar T, G2) is a linear combination oflength oscillations with well known wavenumbers (leftcolumn) and amplitudes with simple dependencies on S-parameters, listed here for the easiest case (10):   1A 00 Þ A00e–2ig1L1 –A 012 Þ±A01e–2ig2L1 –A 022 Þ±A02e–4ig1L1 A012 A11 Þ A11e–4ig2L1 A022 A22 Þ A22e–2i g1 + g2 L1 2A01 A02 A12 Þ±A12Thus the amplitudes and consequently the S-parametersare found explicitly from the solution of an overdetermi-ned system of equations like (10) with G taken from theset of experiments with different L. Although the S-para-meters found in this manner seemed to resemble the mea-sured system behaviour quite well [1,2], consistency testsand experiments simulated numerically (sect. 3) showed,that the S-parameters, especially if weakly contributing,may carry intolerable errors. The reason was found to bethe influence of higher orders of internal reflection, neg-lected so far. Therefore their contribution has to be incor-porated, which is done by iteration. Again, transform (8)using (6), but keep all higher reflection degrees - again asa geometric series - together with G on the left hand side:   G +AT E2 A E2 1 +AE2 –1A E2 A ==A 00 –AT E2 A +AT E2 A E2 A(11)The expression in brackets is easily evaluated for given S-parameters and corrects the measured G for the higherreflection degrees, of course on an approximative base.Then the reduced model on the right is fitted as before tothe left hand side, which gives new S-parameters. Theprocedure is repeated until the variation of the S-para-meters vanishes, which corresponds to the best approxi-mation of the complete model to the measured data. Allconsiderations demonstrated for the calibration step (star-ting from (8)) can be transferred to the measurement  (5).3 SIMULATED EXPERIMENTSIn order to test the procedure a 3-mode set of S-parametersof two adaptors A, B and a device C was generated toevaluate (5) for a typical set of 18 x 18 different522combinations of L1, L2. This was taken as "measured" in-put. Tab. 1 shows 4 (of 21) parameters of C; first the truevalue originally invested, the initial value found by thefit-method and the result of the 4th iteration step. Alt-hough there are significant errors of some of the fit-valuesthe iteration shrinks them within a few iterations dramati-cally. The overall error, defined as the value sum over thedeviations between iteration result and truth of all 21parameters is displayed in Fig. 2 for each iteration step.4 MEASUREMENT SETUP AND EXAMPLESThe various delay line lengths have been realized by buil-ding two adaptor systems sliding in two fixed waveguides.Their design is not critical (beside the supression of trans-verse polarizations) since they have to be calibrated any-how. They are driven by stepping motors with spindlesthat allow for a nominal position resolution of 6.25 mmand a shift of 0.35 m. The RF equipment consists of aHP8753C-6 GHz-network analyzer, connected with theadaptors using phase-stabilized flexible RF cables. Thecomponents are computer controlled. The adaptors and thetest device are insulated to reduce thermal drifts.A typical result of an adaptor calibration is shown in Fig.3. It was found at 4.50 GHz with three propagatingmodes (TE11, TM01, TE21) in a 78 mm diameter circularwaveguide. Fig. 4 demonstrates that the input reflectionG(L) (line) of the adaptor calculated with these S-parameters fits the measured dots very well. Fig. 5 givesan example for results of a device measurement.5 CONCLUSION AND OUTLOOKA method that expands network analyzer functionality inthe regime of multimoded waveguides has been success-fully demonstrated for up to three non-degenerated  propa-gating modes. It provides all information needed todescribe the behaviour of a given waveguide insertion atan arbitrary frequency point (or a spectrum of them). Thusit allows to predict the RF properties of every chain ofknown insertions, which is important not only for acce-lerator sections with respect to (HOM-) fields but for allwaveguide applications with excitation of more than thefundamental mode. Further investigation will be spent inthe resolution of (esp. polarizational) degenerated modesin order to expand the method´s range of applications.SUPPORTWork supported by DESY and BMBF; contract 060f359.REFERENCES[1] H.-W. Glock, P. Hülsmann et. al.: Energy Propagationthrough the TESLA Channel: Measurements with TwoWaveguide Modes (TESLA-Rep. 96-07), DESY, June 1996[2] H.-W. Glock, F. Marhauser et. al.: Measurement of HOM-Propagation through Cavity Chains in Terms of S-Para-meters, Proc. 18th Int. Linear Accelerator Conf., Geneva,26-30 Aug. 1996, pp. 716Parm. Truth First Fit 4th Iterationc[1, 1] 0.1445 + 0.1334 I 0.3185 - 0.2865 I 0.1445 + 0.1334 Ic[2, 2] -0.1162 - 0.1434 I -0.4127 - 0.1202 I -0.1161 - 0.1434 Ic[2, 6] 0.04041 - 0.06209 I 0.04047 - 0.06113 I  0.04041 - 0.06209 Ic[5, 6] 0.2104 - 0.09209 I 0.1841 - 0.1167 I 0.2104 - 0.09192 ITab. 1: Sample S-parameters in simulated experiment2 4 6 8 10 12     -71. 100.000010.0010.110.Fig. 2: Value sum of error of all S-parameters overiteration number. First point is result of fit procedure andused as start value.-Pi -Pi/2 0 Pi/2 PiArg(S-p.)0.10.20.4Abs(S-p.)a00a[1, 1]a[1, 2]a[1, 3]a[2, 2]a[2, 3]a[3, 3]a0[3]a0[2]a0[1]Fig. 3: Typical S-parameter of adaptor, found at 4.5 GHzAbs(G(L))0.1 0.2 0.3 0.4 0.500.20.4Arg(G(L))0.1 0.2 0.3 0.4 0.50Pi2 PiFig. 4: G(L), calculated from the S-parameters in Fig. 3(line) and measured (dots)-Pi -Pi/2 0 Pi/2 PiArg(S-p.)0.20.40.60.81Abs(S-p.)c[1, 1] c[1, 2]c[1, 3]c[1, 4]c[2, 2]c[2, 4]c[3, 3]c[3, 4]c[4, 4]Fig. 5: Example set of S-parameters of a TESLA 9-cellcopper cavity, found at 3.50 GHz with comparably hightransmission in both modes (c[1,3], c[2,4]).523

Analysis of HOM interaction between cavities by multi-modal s-parameter measurements

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