Korsunsky, A M

Repalov, N S

Tur, Yu D

CERN Document Server

Particle Accelerators1980, Vol. II, pp. 15-210031-2460/80/1101-0021 $06.50/0© Gordon and Breach, Science Publishers, Inc.Printed in U.S.A.CONTROL OF THE PARAMETERSOF THE ELECTROMAGNETIC FIELDOF LINEAR ION ACCELERATORSA. M. Korsunsky, N. S. Repalov, and Yu. D. TurPhysico- Technical Institute of the Ukrainian SSR, Kharkov, USSR(Received November 3, 1978; infinal/orm October 8, 1979)The interaction of electromagnetic waves in a cavity loaded with passive resonators representing a stabilization systemfor the accelerating field in a linear accelerator with continuously regulated accelerated particle energies was investigatedtheoretically. The study results in a canonical form of the electromagnetic field in the system of coupled resonators, adispersion equation and shows the particular characteristics of the space distribution of the field and the quality of thesystem.At present, linear ion accelerators are findingincreasing application in industry,1 medicine,2 andmany other sectors ofthe economy.3 This expansionin the range of application of accelerators is leadingto a number of new requirements, most importantamong which is flexibility in energy, Le., the pos-sibility of using the same accelerator for generatingions that are accelerated to various energy levels.Until recently, this problem was solved by succes-sive sections of the accelerator where as manyenergies of the accelerated particles could be ob-tained as there were sections in the accelerator.Continuous energy variation of the ions at the outletwas considered to be generally not feasible. 4 As aresult of a number of studies at KPTI (KharkovPhysical-Technology Institute), however, a methodfor continuous variation of the accelerated ionenergy was developed.5 The method is based on anadditional load on the resonantor from a passive os-cillator. The method most extensively studied andapplied to existing accelerators at present consists ofregulating the ion energy by introducing a post intothe cavity parallel to its axis.6 The load, which isstrongly coupled to the operating and EOII modes,permits a controlled variation of the ratio betweenthe space harmonics of the electromagnetic fieldwith the result that the length of the acceleratingsection is modified. The drawback of this method isthe poor quality of the post oscillator. The presentstudy is of a theoretical investigation of the pos-sibility of regulating the parameters of the electro-magnetic field with a passive load on the operatingresonator with a system of additional high-qualityresonators as proposed by V. A. Bomko et al.?151. CANONICAL FORMULATION OF THEELECTROMAGNETIC FIELD IN A SYSTEMOF COUPLED RESONATORSWe will discuss the natural oscillations of theelectromagnetic field in a system of coupled re-sonators. We will assume that the primary resonatorof the linear ion accelerator Ro is loaded with acertain number of passive cylindrical resonators Rnthrough the coupling apertures SOn (Fig. 1). Thesystem will be studied theoretically according to themethod described in Ref. 8.The Lagrangian of the electromagnetic field in asystem without beam loading is expressed asL = _1_ f(E 2 - H 2 )dV, (1 )8rr vwhere V is the total volume ofthe system and E andH are the electric and magnetic field strengths,respectively.We will define VnO as the region of intersection ofthe field of the primary resonator with that of theload resonator with number n. It is not difficult toshow that Eq. (1) can be expressed asN NL = L o + ~ L m + ~ L nO (2)n =1 n =1whereL = ~ J(E,; - H,;)dV (3)8rr V nL no = -41 1 (En· Eo - H n· Ho)dVrr VnO16 A. M. KORSUNSKY, N. S. REPALOV, AND YU. D. TUR..-------LFIGURE 1 Accelerating system loaded with passive resonators.In the transformation of the Lagrangian into type(2), the electromagnetic field in each load resonatoris represented as a superposition of two fields: thecharacteristic electromagnetic field (En, "n) andthat entering the region ~1O of the electromagneticfield of the primary resonator (Eo, Uo), Le.:E," ==(4)resonator A may be presented in the formA = ~q~ (t) 7T~(r) (5)~On the basis ofthe above hypotheses in respect tothe spectral composition of the electromagneticfield, the field in the primary resonator is calculatedby summation over A, while in the load resonator,the sum consists of only one term corresponding tothe Hili wave.In the primary resonator, the function 'JT\, cor-responding to EOll waves, are represented asand I is the root of the Bessel function I n( x).KA==~R R(6)sin(n¢ + an);K3A ol/Jrr =----.Z,A 2 a 'KA rLrrmzot/J--_.K3A oz 'fzmL 'K3A 1 0 l/J-----K~ r a¢ ,where(The determination of region Wn is clear from Fig.1.) The basis for such a representation of the fieldsis that the operating frequency is of the electro-magnetic field in the accelerator with loading isselected in such a way that space harmonics EOl aremainly excited in the primary resonator. The geo-metric dimensions of the load resonators at anoperating frequency only permit the formation ofHili waves in them. We should mention that such aformulation of the field is possible only when thewavelength corresponding to the operatingfrequency significantly exceeds the characteristicdimensions of the zone of intersection of theresonators.It is useful to take the amplitudes of the vectorpotential A as generalized coordinates. In eachCONTROL OF THE PARAMETERS 17Relation (6) permits us to calculate the part oftheLagrangian describing the electromagnetic field ofthe primary resonator;L == Y2 ~ (Q' 2 - 2 Q2)o 1=0 I WOI I , (7)wherep3/2}L == VVOVn•4/a111sin Kia cos K;Znwhere/ == 0/# 0QI = ( 1;:r2 _W_O_I:_~_:;_a_o)_01 ={ ~d 2 - C2( 2 + 2/2/ 2 .an WOI - Kf 1r L ) IS the natural frequency,EOll is the resonator electromagnetic field mode, R o,V are the radius and volume of the operatIngresonator, respectively, and ao is the first root of theBessel function Jo (x).The Lagrangians of the additional resonators arecalculated analogously and are found to beHere Zn stands for the coordinates of the center ofcoupling aperture n, alII is the first root ofthe Besselfunction J(x), a is the dimension of the couplingaperture in the direction of the Z-axis and P is thearea of the coupling aperture.Therefore, the Lagrangian of the electromagneticfield in the system of coupled resonators has theform2L == Y2 ~ (Q' 2 _ 2/Y.) Q ~ W II =0 I WI'Ll -}L hI,n 8/12where( V) 1/2 (1 _ -2) 1/2Q == _n_ a111h,n 16rr c(8) With this form of the Lagrangian, the generalizedcoordinates are represented by the resolved spectralamplitudes of the electromagnetic field QL and thegeneralized velpcities by their derivatives with re-spect to time QL'Equation (10) together with the Lagrange equations,(9)Whn == c( Of 11 / R~ + rr/L 2n ) 1/2 is the natu r a Ifrequency, and Hill is the mode of electromagneticfield in the resonator with number n.In the calculation of the interaction LagrangianLn,o, we will assume that the geometric dimensionsof the zones of intersection of the primary resonatorwith the additional resonators are small comparedwith the dimensions ofthe resonators. By expandingthe vectors found in the expression for L no in thevicinity of a point that is in the intersecting zone ofthe resonators in a series in (r, qJ) in the primaryresonator and in (p, ~) in the additional resonatorsand by integrating, taking into account the dominantterms, we findL Q i wolnO = -p. hi ~O 0}/2 SlnQI,d aL aLdt aQL - aQL = 0, (11)is fully equivalent to a system of Maxwellequations,9 but with the important advantage ofcontaining only total derivatives.2. DISPERSION CHARACTERISTICS OFTHE LOAD RESONATORWe will study the dispersion characteristics of theelectromagnetic field in the simplest case of twocoupled resonators. For such a system, Eq. (11),taking Eq. (10) into account, assumes the form18 A. M. KORSUNSKY, N. S. REPALOV, AND YU. D. TUR(12 )part permits us to establish the type of wavesinteracting most strongly with the load. When allquantities depend on time as QL == aL exp(iwt), thesystem of equations for the amplitude aL takes theformwhere S, = ~S,. n' The system of equations (12) hasthe integral n (14 )00~J.l, =0Wi8/ 12(15 )a, = J.l ~1/2 (2 2) ah •u, W - WIThe amplitudes a, of the space harmonics of theelectromagnetic field of the primary resonator (onthe basis of the first equation of system 14) areexpressed in terms of the oscillation amplitudes ofthe electromagnetic field in the load resonatorw,S,(13 )expressing the conservation ofenergy ofthe electro-magnetic field. The last term of Eq. (13) representsthe resonator-field coupling energy; analysis of this.", f Mr14l~O709.709 110 11/FIGURE 2 Dependence of the oscillation frequency of theelectromagnetic field on the natural frequency of the passiveresonator in a system of two coupled resonators. Points =experimental data.7CONTROL OF THE PARAMETERS 19By introducing the relation (15) into the secondequation of( 14), we obtain the dispersion law oftheelectromagnetic field in two coupled resonators.By calculating the value of ~wo at the resonancepoint on the basis of Eq. (19) and introducing theresult into Eq. (18), taking (9) into account, weobtain= 1 (16) (20)We note that the infinite sum on the left side ofEq.(16) should be treated as an asymptotic expressionbecause in accordance with the assumption made inSection 1, the system of equations (14) is valid onlyfor I-values substantially smaller thanLIa. It is notdifficult to ascertain that for determination of thenatural frequency of the electromagnetic field in asystem of two coupled resonators, it will be suf-ficient to retain only two terms with indices i and i +1 in Eq. (16), if the operating frequency Wi ~ W ~Wi+ I. In this frequency interval, the dispersionequation of the system takes the formThe calculated dependence of the oscillationfrequency ofthe electromagnetic field in a system oftwo coupled resonators on the natural frequency ofthe HI II mode of the passive resonator is presentedin Fig. 2. Analysis of this figure permits the fol-lowing conclusions: by changing the geometric di-mensions ofthe additional resonator, it is possible totransform the E oIO mode (point M 1) continuouslyinto the E011 mode of the electromagnetic field (pointM 2 ). At the intermediate point M 12 , the electr<rmagnetic field represents a superposition of thespatial distributions of both of these modes. Thedegree of stability of the system in the intermediatestate M I2 may be characterized by the value of thecoupling coefficientKo between wavesE010 andHI I I(21 )whereThe solution of system (21) for Wo 2:: W 2:: WI is2 2 wo- (w - wo)ao + p.-- Soa h = 0V2P == 2ai11/[ao(ai1 - 1)1/2]._Therefore the degree of stability of the system inthe intermediate state is proportional to the thirddegree of the characteristic dimension of the re-sonator coupling aperture.3. REGULATING THEELECTROMAGNETIC-FIELD PROFILEWe will discuss the problem of changing the profileofthe electromagnetic field in the primary resonatorwhen the loading is in the form of one passiveelement tuned in such a way that the naturalfrequency ofthe system is within the limits ofWo< W< WI (section M I , M 2 of the curve in Fig. 2).If the primary resonator is in mode EOIO ' con-tinuous transformation into the E OII mode of theelectromagnetic field occurs; neglecting the influ-ence of higher-order modes, the system ofequations(14) for the field amplitude may therefore be expres-sed as( 17)= 0i 0i +1 (u/ - wh).(18 )The solution ofthe dispersion equation (16) in thevicinity of the resonance point Wo = W h has the formWoao = D( w~ - w2 )Jl V2 Soal == -D( W6 - W 2 )P.W 1S 1(22)where D is an arbitrary constant depending on theenergy of the electromagnetic field in the system.20 A. M. KORSUNSKY, N. S. REPALOV, AND YU. D. TURiasQ6at0.2Or--~~-""----""~-"""'--~~"",,--""""-_--.t'""'"--~~-~_-Q2"0. It-Q6-0.8-jFIGURE 3 Longitudinal field-strength distribution along theaxis of the resonators at different values of the parameter 0 == (w- wa)/( WI - wa).In our approximation, the resultant electro-magnetic field in the operating resonator will takethe form[WO 2 2 2 2E z = Eo VI SO(WI - W ) W1Sl(WO - W ).W [wo 2 2He/> ==- z- Eo r;; SO(WI - W )"0 V 2if Ez == Eo at the point z == r == O.Figure 3 shows the distribution ofthe longitudinalelectric-field strength along the axis of the resonatorfor different values of the ratio of the operatingfrequency of the system to the frequency of modeEolO of the primary resonator. The above data allowus to conclude that under conditions of a continuousincrease of the operating frequency from W o to WI' acontinuous transformation of the electromagneticfield occurs from the profile of mode Eo 10 to that ofmode EOII ' It is this condition that permits con-tinuous regulation of the accelerated-ion energy. Ascan be concluded from Eq. (23), coupling betweenthe fields ofthe primary and additional resonators inthe indicated frequency range increases as theadditional resonator approaches one of the faces ofthe arrangement of primary resonators.4. QUALITY OF THE SYSTEM OF COUPLEDRESONATORSBy loading the linear accelerator with passiveelements with resonance parameters, major changesin all of the characteristics of the electromagneticfield result, making it necessary to clarify theinfluence of such a load on the quality ofthe system.While solving this problem, we will take advantageof the simplifying condition that the geometricdimensions of the additional resonator are, as a rule,CONTROL OF THE PARAMETERS 21substantially smaller than those of the primaryresonator. For this reason, the energy stored in thesystem and the Joule energy losses in the vicinity ofthe operating frequency of the accelerator are com-pletely determined by processes occurring in theprimary resonator. With this stipulation, the qualityof the system can be expressed in the formw J,H2dVQ= -- (24)R s fH 2dswhere Rs is the active part of the surface resistanceof the resonator.By integrating Eq. (24) over the volume and innersurface of the primary resonator, with the use of thethird equation of system (23), we obtain the finalexpression1Q == 2R {S L + 2R o 1wRoL (25)Analysis of Eq. (25) shows that in the frequencyrange Wo< W < WI' the quality ofthe loaded systemappears to be higher than that of the unloadedresonator. The increased quality of the system oftwo coupled resonators was also confirmed experi-mentally.Io It is clear that this phenomenon dependson the partial extraction of the electromagnetic fieldfrom one ofthe faces ofthe primary resonator and onan increase in operating frequency.In summarizing the results obtained, we wouldlike to indicate that because of the great importanceof the coupling coefficients between the naturaloscillations of the intersecting resonators and theirhigh total quality, a system ofthis type appears to bevery promising for the development of linear ac-celerators with continuously regulated ion energy.REFERENCES[1] N. S. Solovyev, Atomnaya Tekhnika Zarubezham 12, 33(1973).[2] V. A. Glukhikh, Proc. 5th USSR National Particle Ac-celerators, II, Nauka 1977, p. 47.[3] N. S. Solovyev, Atomnaya Tekhnika Zarubezham 5, 42(1974).[4] A. V. Babushkin, B. I. Bondarev, and L. Yu. So}ovyev, Proc.3rd USSR National Conf. on Particle Accelerators, II,Nauka 1973, p. 258.[5] V. A. Bomko, British Patent No.1, 311,616, Bull.Izobreteniya zarubezhom, No.6, 1973.[6] V. A. Bomko et aI., Particle Accelerators 3, 63 (1972).[7] V. A. Bomko et aI., 5th USSR National ConL on ParticleAccelerators, I, Nauka 1977, p. 337.[8] N. S. Repalov, Particle Accelerators 8, 81 (1978).[9] G. Wentzel, Introduction to Quantum Theory ofWave Fields(Gostekhizdat 1947), p. 146.[lO]V. A. Bomko and B. I. Rudyak, Voprosy Atomnoy Nauki iTekhniki. Fizika Vysokikh Energii i Atomnogo Yadra[Issues of Atomic Science and Technology. Particle Physicsand the Atomic Nucleus] 2( 13), 8 (Kharkov) 1975.

Control of the parameters of the electromagnetic field of linear ion accelerators

Abstract

Similar works

Full text

Available Versions

CERN Document Server