78 Wei Yan-Yu et al. Vol. 11 U 0 (k 1 ) = J v(k 1 c )J 0 v(k 1 ) J v (k 1 c )J 0 v(k 1 ) J v (k 1 c )J v (k 1 b ) J v (k 1 c )J v (k 1 b ) ; V (k) = J

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1 Vol 11 No 3, Mach 00 cfl 00 Chin. Phys. Soc /00/11(03)/ Chinese Physics and IOP Publishing Ltd Dielectic eect on the f chaacteistics of a helical goove tavelling wave tube * Wei Yan-Yu( Ω) a)y, Wang Wen-Xiang(±ΦΨ) a), Sun Jia-Hong(Π ) a), Liu Sheng-Gang(ΛΞ ) a), Baofu Jia a)b), and Gun-Sik Pak b) a) National Key Laboatoy of High Powe Vacuum Electonics, Univesity of Electonic Science and Technology of China, Chengdu , China b) Vacuum Electophysical National Laboatoy, Depatment of Physics, Seoul National Univesity, Koea (Received 14 Febuay 001; evised manuscipt eceived Novembe 001) A new type of patial-dielectic-loaded helical goove slow-wave stuctue (SWS) fo millimete wave tavelling wave tube (TWT) is pesented in this pape. The adio-fequency chaacteistics including the dispesion popeties, the longitudinal electic field distibution and the beam-wave coupling impedance of this stuctue ae analysed. The esults show that the dispesion of the helical goove cicuit is weakened, the phase velocity is educed and the position of the maximum E z is moved fom the mouth to the inside of the goove afte patially filling the dielectic mateials in the helical goove SWS. Theefoe, the dielectic-loaded helical goove SWS is suitable fo a multi-beam TWT with boad band and high gain. Keywods: dielectic-loading, helical goove waveguide, slow wave stuctue, tavelling wave tube PACC: 7170G, 7145L, Intoduction Duing the development of a high-powe boad band tavelling wave tube (TWT), attention has been newly focused on the whole-metal slow-wave stuctues (SWS), [1] such as the helical goove waveguide, [] coaxial adial line, sepentine waveguide and ß-type line, etc, due to thei supe powe capability and potential boad bandwidth chaacteistics. The helical goove waveguide is one of the most impotant wholemetal SWS fo millimete wave TWT. Thee ae seveal peculia chaacteistics, such as the lage tansvese size, the high pecision of manufactuing and assembly, the pefect wholeness and stability in the stuctue. The helical goove waveguide was fist investigated as a SWS in 1957 by Waldon. [3] Late, Nwachuku [4] and Foulds and Mansell [5] deliveed contibutions to the cicuit. In 1988, the fist expeimental idge-loaded helical goove millimete wave TWT was caied out at the Raytheon Company, [6] and the theoetical study was pesented by the authos in [] Pevious woks [ 7] have shown that the bandwidth of the helical goove waveguide is naow. The instantaneous bandwidth of Raytheon's tube is only 1.3%. [6] Theefoe, it is necessay to boaden the band of this stuctue. Although much wok has been done on the dieent types of helical goove waveguide, [8;9] an investigation of the dielectic-loaded helical goove waveguide, as fa as we know, has not appeaed in the liteatue. Recently, the poblems associated with dielectic-loaded TWT fo the advanced acceleato have become moe attactive. [10;11] In this pape, we study the patial-dielectic-loaded helical goove waveguide as a slow-wave cicuit fo TWT. By patially filling the goove with dielectic mateials, the dispesion of the SWS becomes weak, i.e. thee is a small phase velocity change of the electomagnetic wave in the patial-dielectic-loaded helical goove waveguide within a boad fequency egion. This leads to an eective inteaction between the beam and slow waves in the cicuit with a elatively wide band. Afte patially filling the helical goove SWS with dielectic mateials, the position of the maximum E z is changed fom the mouth to the inside of the goove. To incease the inteaction impedance, seveal holes ae dilled on the inside wall of the goove and the beam is put though the holes. Theefoe, the patially-dielectic-loaded helical goove SWS is suitable fo multi-beam TWT. Λ Poject suppoted by the Fund of the Institute of Electonics Science of the Ministy of Electon Industy (Gant No DJ3.1.1.). y yywei@uestc.edu.cn; Phone: ; Fax:

2 78 Wei Yan-Yu et al. Vol. 11 U 0 (k 1 ) = J v(k 1 c )J 0 v(k 1 ) J v (k 1 c )J 0 v(k 1 ) J v (k 1 c )J v (k 1 b ) J v (k 1 c )J v (k 1 b ) ; V (k) = J v(k 0 ) G 0 J v (k 0 ) J v (k 0 b ) G 0 J v (k 0 b ) ; V 0 (k) = J0 v(k 0 ) G 0 J 0 v(k 0 ) J v (k 0 b ) G 0 J v (k 0 b ) ; k 1 =! p " " 0 μ 0 = k 0 p " ; k 0 =! p " 0 μ 0 : Fig.1. The longitudinal coss section of the patiallydielectic-loaded helical goove waveguide..dispesion and impedance.1. Field components Figue 1 shows the longitudinal section though the axis of the patially-dielectic-loaded helical goove slow-wave cicuit. The goove egion is patially filled with lossless dielectic mateials fom b to c. The elative pemittivity of the mateial is ". In Fig.1, p and w epesent the pitch and goove beadth, espectively, and c, b and a indicate the oute adii of egions I, II and III, espectively. In the following analysis, the pitch is assumed to be suiciently small compaed with the wavelength so that the high-ode evanescent modes in the goove egions I and II may be neglected. Fo the field expessions obtained by employing the appoximate helical coodinate system (; ;ο) [5] (one may also use the developed helical coodinate system [1] ), the field components of the fundamental TE mode in the helical goove waveguide can be witten as follows. In egion I: ( b»» c ) Hee E I ο = AU(k 1 )e jv ; H I = (v=!μ 0 )AU(k 1 )e jv ; H I = j(k 1 =!μ 0 )AU 0 (k 1 )e jv ; E I = E I = H I ο = 0: In egion II: ( a»» b ) (1a) (1b) (1c) (1d) ο = BV (k)e jv ; (a) = (v=!μ 0 )BV (k)e jv ; (b) = j(k 1 =!μ 0 )BV 0 (k)e jv ; (c) = = ο = 0: (d) U(k 1 ) = J v(k 1 c )J v (k 1 ) J v (k 1 c )J v (k 1 ) J v (k 1 c )J v (k 1 b ) J v (k 1 c )J v (k 1 b ) ; A, B and G 0 ae the field amplitude coeicients, J ±v (x) epesents the Bessel function of fist kind with ode ±v, and J 0 ±v(x) epesents the fist deivative of J ±v (x) with espect to x. k 0 is the fee-space popagation constant, and v is the angula popagation coeicient fo waves popagating ound the goove. It can be detemined by the fact that the phase shift pe pitch in the cente egion should be equal to the phase change pe tun in the goove, i.e. v = fi 0 p=ß. Hee fi 0 is the axial phase popagation constant in the cente egion. The field components in the cente egion can be obtained by the Floquet theoem I z = I z I i = ± j = j i C m fl m F m(fl m )e j(mi fimz) ; [C m fl m fi m Fm(fl 0 m ) (3a) D m (m!μ 0 =)F m (fl m )]e j(mi fimz) ; (3b) = [C m (mfi m =)F m (fl m ) D m!μ 0 fl m F 0 m(fl m )]e j(mi fimz) ; (3c) D m fl m F m(fl m )e j(mi fimz) ; = ± [C m (mk =!μ 0 )F m (fl m ) = ± j (3d) D m fl m fi m F 0 m(fl m )]e j(mi fimz) ; (3e) [C m (k fl m =!μ 0 )Fm(fl 0 m ) D m (mfi m =)F m (fl m )]e j(mi fimz) : (3f) Hee C m and D m ae the field amplitude factos of the mth space hamonic. fi m and fl m ae its axial

3 No. 3 Dielectic eect on the f chaacteistics of a helical goove and adial popagation constants, espectively. They ae defined as fi m = fi 0 + mß=p, flm = jfim k j. If fim k > 0, F m (fl m ) = I m (fl n ) and the uppe sign of ±" o " is selected; if k fim > 0, F m (fl m ) = J m (fl m ) and the lowe sign of ±" o " is chosen. J m and I m epesent Bessel and modified Bessel functions of ode m, espectively... Dispesion equation At = b, the tangential E-fields and H-fields ae continuous E I ο = ο ; (4) H I = ; (5) while at = a, the matching conditions of E-fields ae ο = I ο ; (6) = I = 0: (7) H -fields should be continuous at the bounday acoss the goove Z (N +1)p Np dz = Z (N +1)p Np dz: (8) Fom Eqs.(4) (8), the following dispesion equation of the patially-dielectic-loaded helical goove SWS can be obtained kp v (k 1 c ;k 1 b )T v (k b ;k a )+k 1 Q v (k 1 c ;k 1 b )R v (k a ;k b ) kp v (k 1 c ;k 1 b )R v (k b ;k a )+k 1 Q v (k 1 c ;k 1 b )P v (k a ;k b )» ± w k F 0 m (fl m a ) 1 p fl m F m (fl m a ) F m (fl m a ) mfim kfl m Fm(fl 0 fl m a ) mtg Ψ a sin fim w= = 0; fi m w= (9) whee the functions P v (x; y), Q v (x; y), R v (x; y) and T v (x; y) ae defined as follows P v (x; y) = J v (x)j v (y) J v (x)j v (y); Q v (x; y) = J v (x)j 0 v(y) J v (x)j 0 v(y); R v (x; y) = J 0 v(x)j v (y) J 0 v(x)j v (y); T v (x; y) = J 0 v(x)j 0 v(y) J 0 v(x)j 0 v(y): J v (k 0 ) should be eplaced by Y n (k 0 ) in the above equations when v is an intege n. Fo the device without dielectic-loading, i.e. " =1, Eq.(9) can be completely identified with the esults given by Foulds and Mansell [5]..3. Coupling impedance The coupling impedance is an impotant paamete that descibes how stong the beam wave inteaction is in a TWT. Fom Piece's theoy, [13] the coupling impedance of the mth space hamonic is defined as K c (m) = E zme Λ zm fi m P ; (10) whee E zm is the longitudinal component of the electic field of the mth space hamonic at the position e of the electonic beam, and E Λ zm is its conjugate value. Accoding to Eq.(3a) and the elation between C m and B, we have E zm E Λ w zm = p B» J v (k a ) G 0 J v (k a ) F m(fl m e ) F m(fl m a ) J v (k b ) G 0 J v (k b )» sin(fim w=) : (11) fi m w= In Eq.(10), P isthe total powe flow though the whole cicuit system P = P I + P II + P III m ; (1) whee P I and P II ae the powe flows in egions I and II, espectively. Pm III is the powe flow of the mth space hamonic in space III. The expessions of P I, P II and P III ae P I = 1 w Z c b E I ο HΛI d Z = 1 v c d wa!μ 0 b» Jv (k 1 c )J v (k) J v (k 1 c )J v (k) : J v (k 1 c )J v (k b ) J v (k 1 c )J v (k b ) Afte evaluating this new type of integal, the expes-

4 80 Wei Yan-Yu et al. Vol. 11 sion becomes P I = wa 1 4!μ 0 [J v (k 1 c )J v (k 1 b ) J v (k 1 c )J v (k 1 b )] k 1 b [J v (k 1 c )J (v+1) (k 1 b )+J v (k 1 c ) J v+1 (k 1 [J v(k 1 c )J v (k 1 b ) J v (k 1 c )J v (k 1 b )] [J v (k 1 c )J v (k 1 b ) J v (k 1 c )J v (k 1 [J v(k 1 c )J (v+1) (k 1 b ) + J v (k 1 c )J v+1 (k 1 b )] [J v (k 1 c )J v (k 1 b ) J v (k 1 c )J v (k 1 b )] : (13) Similaly P II = 1 w Z b a Eο II HΛII d = wb 1 4!μ 0 [J v (k b ) G 0 J v (k b )] k [J (v+1) (k) + G 0 J v+1 (k)] the tem numbe m inceases. Fo example, only fou tems (m =, 1, 0, 1) ae calculated, and the elative eo of the solution may be less than In ou calculation, we have taken seven tems into account (jmj» 3). Figue shows the dispesion chaacteistics of the fundamental wave in the dielectic-loaded helical goove SWS fo c =p=6.6, b =p=4., a =p=3., w=p=0.5 and " =3.8. As one would expect with a closed SWS, the helical goove stuctue exhibits a cut-o fequency, and can popagate fast waves and slow waves (Fig.(a)). Moeove, it can be seen fom Fig.(b) that the cuve of phase velocity vesus fequency of the dielecticloaded helical goove stuctue becomes flatte than that of the countepat without dielectic-loading. This indicates that the beam-wave inteaction will be maintained in a elatively wide fequency egion in the fome case. In othe wods, we can obtain a boade band amplifie in the dielectic-loaded helical goove [J v(k) G 0 J v (k)] [J v (k) G 0 J [J (v+1)(k)+g 0 J v+1 (k)] [J v (k) G 0 J v (k)] =a Z a = b ; (14) Pm II = 1 [E III HΛIII i Ei III HΛIII ]ßd 0 w sin(fim w=) fim =ß a [BV (k a )] p fi m w=!μ 0» 1 k mfim + 4 fl m flm tan Ψ a» Fm (fl m a ) F m+1 (fl m a )+Fm 1(fl m a ) Fm(fl 0 m a ) Fm(fl m a ) F m+(fl m a )+F m (fl m a ) F m (fl m a ) m (fi m + k ) fl m [(mfi m =flm a) tan Ψ] F m (fl m a ) fl m a fi m a Fm(fl 0 m a ) (15) : 3. Numeical esults and discussion The dispesion equation (9) is a vey complicated tanscendental equation including integals associated with Bessel functions and the summations of an infinite seies. Fotunately, the seies conveges apidly as Fig.. Dispesive chaacteistics of the dielectic-loaded helical goove SWS fo c=p=6.6, b =p=4., a=p=3., w=p=0.5, " =4.0: (a)! c=c-fi 0 p cuves; (b) v p=c-! c=c cuves.

5 No. 3 Dielectic eect on the f chaacteistics of a helical goove The influence of the dielectic-loading on the coupling impedance is illustated in Fig.3, in which the physical paametes ae same as those in Fig.. Hee, the adius of the beam e is assumed to be 0.9 a. It is shown that the dielectic-loading educes the coupling impedance of the fundamental wave. This decease of coupling impedance is not desiable because it will cause a eduction of the gain and the eiciency of the tube. Theefoe, the value of the elative pemittivity of the dielectic should be popely chosen. hollow beam nea a may be employed fo high eiciency of the beam-wave inteaction. We also obseve that, when the stuctue is loaded with dielectic, the position of the maximal E z is moved to the goove space (Fig.4). This is due to the fact that the loaded dielectic makes the electomagnetic enegy concentate inside the goove. [14] Then, the beam-hole position should be selected in the goove egion to obtain high gain and high eiciency. In this case, multibeams may be employed. Fig.3. Coupling impedance of the dielectic-loaded stuctue (the stuctue paametes ae the same as those in Fig.). In ode to mitigate the eect of the dielecticloading on coupling impedance, we examine the distibution of the longitudinal electic field E z along the diection, which eveals the natue of the beam-wave inteaction. In Fig.4, the pofiles of E z ae shown fo seveal values of dielectic constant ". The position maked by the dashed line is = a. It is evident fom Fig.4 that the field amplitude inceases apidly with the incease of, and at = a it eaches an extemum fo the case without dielectic-loading. Thus, in the geneal case without dielectic-loading, a thin Fig.4. The longitudinal electic field pofiles fo seveal values of dielectic pemittivity. 4. Conclusion In this pape, we have pesented a patialdielectic-loaded helical goove waveguide as a slowwave stuctue fo a tavelling wave tube and we put the emphasis on the eect of the dielectic paametes on the f chaacteistics of the stuctue. It is shown that the dielectic-loading can make the stuctue less dispesive, but educe the coupling impedance. In ode to incease the gain and output powe of the tube, the use of this type of stuctue with multi-beam holes lying at the goove wall is suggested. Refeences [1] Epsztein B 1984 IEDM'84 M486 [] Wang W X, Yu G F and Wei Y Y 1997 IEEE Tans. Micow. Theoy Tech [3] Waldon R A 1957 J. B. IRE [4] Nwachuku M A 1961 PhD Dissetation Depatment of Electical Engineeing, Univesity of London [5] Foulds K W H and Mansell J R 1964 Poc. IEE [6] Liss C, Hape R and Pui M P 1988 IEDM'88 M374 [7] Wei Y Y, Wang W X et al 000 Acta Phys. Sin (in Chinese) [8] Wang W X, Lan Y H and Wei Y Y 000 Int. J. Infaed Millim. Waves [9] Wei Y Y, Wang W X and Sun J H 000 IEEE Micow. Guid. Wave Lett [10] Gai W 000 Conf. Digest of 5th Int. Conf. on Infaed and Millimete Waves (Beijing, China) M61 [11] Gai W, Conde M, Konecy R et al 1998 Poc Paticle Acceleato Confeence, IEEE 1 M633 [1] Wang Z, Chen Q and Wu H 1999 Acta Electon. Sin (in Chinese) [13] Piece J R 1950 Taveling Wave Tubes (Pinceton, NJ: Van Nostand) [14] Qain B, Li C L and Liu Y G 1996 Phys. Rev. E