Real Part of the Impedance for a Smooth Taper*

Transcription

1 SLAC-PUB Ocober 995 Real Par of he Ipedance for a Sooh Taper* G. V. Supakov Sanford Linear Acceleraor Cener Sanford Universiy, P.O. Box 4349, Sanford, CA 9439 Absrac Real par of he ransverse and longiudinal ipedance of a pipe wih slowly varying radius is calculaed based on he energy radiaed by he bea in he ransiion region. Subied for publicaion Work suppored by Deparen of Energy conrac DE-AC3-76SF55

2 I. INTRODUCTION In his paper we calculae he real par of he ipedance for a sooh axisyeric aper whose radius slowly varies along he axis. Previously Yokoya found a purely iaginary longiudinal and ransverse ipedances for such srucures []. This, however, does no ean ha he real par of he ipedance vanishes idenically; i only indicaes ha, in he doain of validiy of Yokoya's resul, i is uch saller han he iaginary par. For and Re Z l ( ω) have been found in a sall variaion b of he pipe radius b, boh I Z l ω Ref. []. We generalie he resul of Ref. [] for arbirary b, bu lii our consideraion o he real par of he ipedance only. The ehod used in his paper is based on he WKB heory of waveguide odes in sooh srucures. General reaen of WKB approxiaion for Waveguides can be found in Ref. [3]. We calculae he apliudes of waves generaed by he bea in he aper and relae he radiaed energy o he real par of he ipedance. A siilar approach was previously used for he calculaion of he real par of he ipedance of slos and holes in a vacuu chaber [4]. We presen a deail derivaion of he real par of he longiudinal ipedance and discuss applicabiliy liis of he heory. The derivaion of he ransverse ipedance can be perfored in he sae fashion; in his paper we will give he final resuls oiing he lenghy derivaion. II. RADIATION CAUSED BY THE TANGENTIAL ELECTRIC FIELD AT THE WALL We begin our consideraion wih a sraigh circular perfecly conducing pipe of consan radius b wih a localied volage Vexp( iω ) applied o he wall a he locaion = and oscillaing wih he frequency ω. Maheaically, his siuaion corresponds o he following boundary condiion, E = Vδ, () r= b where is he coordinae along he axis of he pipe, and we oi he facor exp( iω) hroughou his paper. Equaion () reads ha he angenial elecric field a he wall vanishes everywhere excep he poin = where i has a singulariy such ha Ed = V. The soluion of he Maxwell equaions inside he pipe wih he boundary condiion () can be found in ers of he axisyeric TM odes:

3 E jb J j r = i () exp b ( σϕ ), ijκ r Er = σ J j exp i () b b ( σϕ ), () ω H ijbc J j r = θ exp i () b ( σϕ ), where J and J are he Bessel funcions of he eroh and firs order, j is he h roo of J, b is he radius of he waveguide, ϕ()= κ, κ = ω ω c, where ω = cj b is he cuoff frequency for he h ode. The variable σ denoes he direcion of agaion of he wave; σ =+ corresponds o he odes againg in he posiive direcion along he -axis, and σ = arks he waves raveling in he opposie direcion. Below he cuoff frequency, ω < ω, κ is purely iaginary wih a posiive iaginary par Iκ >. The elecroagneic field generaed by he volage V in Eq. () can be found using he resuls of Ref. [5]. I is given by he cobinaion of he TM odes wih he apliudes a againg in he posiive and negaive direcions, = ( + ) ( ) F = h a F ( r,, σ )+ h + a F ( r,, σ ), (3) = = () is he sep funcion, F denoes any of he coponens E, E r, or H θ, and a + where h ( and a ) refer o he waves againg in he posiive and negaive direcions, respecively. For he apliudes a one finds, ( σ iv a ) = exp ( σκ i ). (4) κjj( j) As an illusraion, in Fig. we plo he elecric field E on he axis for wo differen values of he frequency ω. I is clear ha he soluion given by Eqs. (3) and (4) plays a role of a Green funcion and allows us o find he fields for an arbirary axisyeric disribuion of he angenial elecric field a he wall. = III. A BEAM IN A PIPE WITH A VARYING RADIUS We now consider a relaivisic bea againg in he pipe wih a gradually varying radius b (), as shown in Fig. 3. We assue ha he angle beween he wall of he pipe and he axis is sall, b <<. The bea will be represened by an oscillaing curren raveling wih he speed of ligh along he axis of he pipe, where k =ω c. = I, I exp ik, (5) 3

4 .8 E b/v /b Fig.. Elecric field E on he axis for ωb c=. The field exponenially decays away fro he source because he frequency ω is below he cuoff frequency. 5 E b/v /b Fig.. Elecric field E on he axis for ωb c=. Two waveguide odes, TM and TM, are excied and agae away fro he source in posiive and negaive direcions. 4

5 Fig. 3. A sooh aper. The elecroagneic field generaed by he bea in he pipe, Er (), Hr (), can be vac represened as a su of he field ha he bea has in he vacuu, E (), r plus he radiaion field produced by he currens in he pipe wall, Ẽ(), r vac Er ()= E ()+ r Er (), (6) vac (and a siilar expression for he agneic field). The elecric field E () r has a radial ( vac coponen a he wall which agniude is equal o E ) r = [ I b() c] exp ( ik). Because he surface of he wall has an angle b wih he -axis, his field gives rise o he angenial coponen a he wall, I E vac = cb b exp ( ik ). (7) In order o saisfy he boundary condiion a he perfec conducor, E =, he angenial coponen of he radiaion field a he wall us be equal o E vac, vac I E E exp cb b ik = =. (8) The proble hus reduces o finding he soluion of he Maxwell equaions subjec o he boundary condiion given by Eq. (8). We will find his soluion in he firs approxiaion o he sall paraeer b using he Green funcion derived in Secion II. IV. SHALLOW TAPER Before addressing a general proble of arbirary cross funcion b (), we consider firs a shallow aper for which he change of he pipe radius is sall, ()= + () << b b b, b b. (9) 5

6 Applying he boundary condiion (8) o his case, in he firs approxiaion, we can neglec he variaion of he pipe radius and require ha Eq. (8) be saisfied a he surface of he pipe wih a radius equal o b. The soluion can be easily found wih he use of he Green funcion fro Secion I, where E σ =± [ ] E ()= r ( + ) () E ( ) ( ( r, = )+ ) c σ c () E ( r, σ = ), () is he elecric field fro Eq. (), and c c ( + ) ( ) ()= κ ()= κ i j J j i j J j E exp iκ d, E exp iκ d. ( ) ( ) Having found he elecric field in he pipe, one can ry o find he longiudinal ipedance using he definiion Zl = de( r = ) ( ik) I, exp. () This will however give a rivial resul Z Zl = [ b b( ) ], (3) πb which is due o he change of he cross secion of he pipe and vanishes in he case of a colliaor for which b= b( ). I urns ou ha a nonrivial par of he ipedance is orional o b, and a sraighforward way o find i would be o develop a second order of he perurbaion heory for he elecroagneic field, and hen o use Eq. (). We, however, choose a sipler approach ha, hough does no allow o find he iaginary par of he ipedance, will give us Re Z l wihou going o he second order heory (cf. Ref. [4]). The approach is based on he exac relaion beween he power P radiaed ino he waveguide due o he presence of he aper and he real par of he ipedance, In our case, he radiaed power is = () P I Re Z ω. (4) = ( + ( = ) ) ( TM) + P = P c = c, (5) TM where P 8 ωκ j J j is he energy flow in he TM ode of uni apliude, and he su carries over he againg odes only. The apliudes of ougoing waves a ± are c ( ± ) ii ( =±)= cbκ j J j bˆ k κ, (6) 6

7 where he cap denoes he Fourier ransfor, fˆ ( k)= df () exp( ik). (7) This gives he following resul for he ipedance, Re Z ( Zk bˆ k bˆ lω )= k πb κ ( κ ) + ( + κ ). (8) 4 For any given frequency, he conribuion o Re Z l coes only fro he finie nuber of odes whose cuoff frequencies are ω < ω. The ipedance (8) agrees copleely wih he Warnock resul []. V. WKB APPROXIMATION IN A WAVEGUIDE WITH A SLOWLY VARYING RADIUS To lif he assupion of he shallowness of he aper we need o consider he agaion of he eigenodes in a waveguide wih a varying radius. In general, for an arbirary funcion b (), he elecroagneic waves given by Eq. () do no provide a correc represenaion of he eigenodes in his case. A TM ode inciden on he aper fro a sraigh pipe will experience a ransforaion o oher odes and a possible ecion if he cuoff frequency a he salles cross secion of he aper is higher han he frequency of he ode. However, if he ransiion is sooh and he conversion ino oher odes can be negleced, he paraeers of he againg ode adiabaically adjus o he local value of he radius and he ode keeps is ideniy. The agaion of he ode in his case can be described wihin he fraework of he WKB approxiaion which is valid if he inverse wavenuber of he ode, κ, is uch saller han he characerisic lengh l of he aper. Typically, κ < b, and he condiion κ << l is auoaically saisfied for b<< l. Only for odes ha are close o he cuoff frequency, corresponding o κ, he WKB approxiaion fails and a ore accurae reaen is needed. In he WKB approxiaion, he longiudinal wavenuber of he ode, κ, is a funcion of posiion, κ ()= k and he corresponding phase ϕ is given by j b (), (9) ϕ ()= κ d, () 7 where he lower lii can be chosen arbirarily. The cuoff frequency of he ode is also a funcion of he posiion, ω ()= cj b(). The srucure of he ode is obained fro Eq. () by subsiuing b() for b, and adding an apliude facor a () which is also a funcion of (for he sake of breviy, we wrie here only he equaion for he -coponen of he elecric field),

8 ( j E ) a b J j r = () i () exp( σϕ ). () () b () The apliude a () urns ou o be inversely orional o he square roo of he longiudinal wavenuber κ [3], cons a ()=. () κ () This dependence can be easily undersood fro he fac ha he power flow in he wave, P a κ, us be consan a each cross secion of he pipe. The above equaions are applicable in he region where ω > ω(). When a wave approaches a urning poin r a which ω = ω( r), i will be eced in he opposie direcion. As a resul of he ecion, he wave ges a phase advance π [3]. We can wrie down he correspondence beween he inciden and eced waves in he following for () a () a j b J j r b j b J j r b exp[ iσϕ ( () ϕ ( ))] r π exp iσϕ ( () ϕ ( )) i. r The above consideraion assues ha he wave againg in he waveguide wih varying cross secion changes is apliude bu does no ransfor ino he oher odes. This ipose an iporan requireen on he soohness of he aper. Naely, in order o be able o neglec he ransforaion effecs, he lengh of he ransiion region should saisfy he following condiion [3] kb << l, (4) (we assue here ha variaion of he radius b is of he order of b). For a given aper, Eq. (4) will deerine he frequency range for which he resuls of his paper will be applicable. VI. GREEN FUNCTION IN WKB APPROXIMATION AND LONGITUDINAL IMPEDANCE We are now in a posiion o derive a Green funcion corresponding o he soluion of he Maxwell equaions for a long aper wih he boundary condiion given by Eq. (). The resul can be obained fro he following arguen. Because of he soohness of he pipe, he apliude of he TM odes radiaed by he source () in is viciniy (ha is a he disances of several local radii, bu uch saller han he lengh of he aper) is given by Eq. (4) in which κ = κ is he local value of he wave nuber. Traveling fro he poin o he poin, he wave will change is phase and apliude in accordance wih Eqs. () and (). This resuls in he following expression for he apliude of he wave a a he posiion,, (3) 8

9 [ ] iv a(, )= exp σi( ϕ() ϕ ). (5) κ() κ jj( j) We can now repea he arguens of Secion III and find ha he radiaion of he bea (5) is given by he sae Eq. () wih he apliudes ( ± ii c ) b ()= ζ d ik i( ( )) cb () j J ( j ) ζ exp[ ζ ϕ ζ ϕ ](6) κ b ζ κ ζ (we reind ha in Eq. () one now has o use he phase ϕ () fro Eq. () raher han ( + κ ). The inegraion in Eq. (6) runs fro o for c ) (, and fro o for c ). Using Eq. (6) we can find he energy radiaed by he bea in he ransiion region, Eq. (5), and he real par of he ipedance, Eq. (4). The resul is Re Z l ( ω)= + Zk 4π b ζ dζ b ζ κ ζ b ζ dζ bζ κ ζ [ ] exp ikζ iϕ ζ [ + ] exp ikζ iϕ ζ. The superscrip "" indicaes ha he ipedance is due o he odes ha agae rough he aper, ha is in( b ())> j k. There ay also be odes ha are eced by he aper, for which in[ b ()]< jo k< ax[ b ()]. Conribuion of hose odes is given by he following expression, r Zk b ζ π Re Zl ( ω)= dζ exp( ikζ) cos ϕ ζ ϕ r π + b( ζ) κ ( ζ) 4 The oal ipedance Re Z l is he su of Eqs. (7) and (8), Re Z ω Re Z ω Re Z ω l l l (7). (8) = +. (9) VII. REAL PART OF THE TRANSVERSE IMPEDANCE Calculaion of he real par of he ransverse ipedance can be perfored siilar o he longiudinal one. One has o find he energy of he dipole TE and TM odes radiaed by he bea ino he pipe while passing hrough he aper. Here we wrie down he expressions for Re Z wihou derivaion. Firs, we need o define a local wavenuber for TM and TE odes againg in a pipe wih varying radius, ()= = () TM TE k k j b, k k j b, (3) where j and j are he h roos of he Bessel funcion J and is derivaive J respecively. Inroducing he phase of a ode as an inegral of he wavenuber, 9

10 ( TM) ( TM) ϕ k ζ dζ, (3) ()= ( and a siilar expression for ϕ TE ) (), we can wrie he real par of he ipedance in he following for, where and Re Z ω Z ω Z ω = +, (3) = Z ω ZTE + ω ZTE ω ZTM + ω ZTM ω, (33) = +. (34) Z ω ZTE ω ZTM ω Each er in Eqs. (33) and (34) is associaed wih he energy radiaed ino TE or TM odes againg in he posiive (+) or negaive (-) direcions. The superscrips "" and "" refer o he odes ha can eiher agae hrough he aper or will be eced back due o he increase of he cuoff frequency in he narrow par of he aper. The expressions for each er in Eqs. (33) and (34) are Z Z () Z TE ± ( ω)= π b d b k e TE () Z TM ± ( ω)= π j b d b k e TM ( TE ) ik± iϕ, (35) ( TM ) ik± iϕ, (36) Z TE ( ω)= Z π ± r () b d b k e ik ( TE) ( TE) ( TE) π cos ϕ TE () ϕ ( r ), (37) 4 Z TM ± Z b d j b k e ik TM TM TM π ( ω)= ϕ TM () ϕ ( r ) π cos.(38) 4 r () ( TE) ( TM) ( TM where r and r are he poins a which k ) ( TE = and k ) =, respecively. The suaion in Eqs. (35) and (36) carries over he odes for which in( b ())> j k, and in( b ())> j k, respecively. Siilarly, he suaion in Eqs. (37) and (38) is perfored over he odes for which in[ b ()]< j k< ax[ b ()], and in[ b ()]< j k< ax[ b ()], respecively. The evanescen odes, such ha j k, j k > ax[ b() ] do no conribue o he real par of he ipedance. In he lii of a shallow aper considered in Secion IV, Re Z can be siplified. ( Firs, he conribuion of he eced odes can be negleced. Second, ϕ TE ) () and

11 ( ϕ TM ) ( TE) ( TM) ( TE) ( TM) () can be subsiued by k and k, respecively, where he k and k can be considered as no depending on. Finally, b () can replaced by b. The resul Z Re Z( ω)= 4 ( TM) πb k ( j ) agrees wih [6]. db e ( ik ik db e TE ) + ( d TE) () + k b () e ( ik ik TM ) ( TM ) + ik ik () + db () e ( TE ) ik ik, + (39) VIII. ACKNOWLEDGMENT I would like o hank R. Warnock for useful discussions. References. K. Yokoya. Ipedance of Slowly Tapered Srucures. CERN SL/9-88 (AP) (99).. R. L. Warnock. An Inegro-Algebraic Equaion for High Frequency Wake Fields in a Tube wih Soohly Varying Radius. SLAC-PUB-638 (993). 3. B.Z. Kasenelenbau. Theory of Nonunifor Waveguides wih Slowly Varying Paraeers. Moscow: Id. Akad. Nauk SSSR, 96 (in Russian). 4. G.V. Supakov. Puping Ipedance of a Long Slo and an Array of Slos in a Circular Vacuu Chaber. Physical Review E, vol. 5, p. 355 (995); SLAC-PUB L.A. Weinsein. Elecroagneic Waves (Radio i svya', Moscow, 988). 6. R.L. Warnock, privae counicaion.