Mechanism analysis of radiation generated by the beam-plasma interaction in a vacuum diode *

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1 06 Hfi Institts of Physical Scinc, Chins Acady of Scincs and IOP Pblishing Printd in China and th UK Plasa Scinc and Tchnology (9pp) doi:0.088/ /9//05003 Mchanis analysis of radiation gnratd by th baplasa intraction in a vac diod * Zngchao JI ( 季曾超 ), Shixi CHN ( 陈仕修 ) and Shn GAO ( 高深 ) School of lctrical nginring, Whan Univrsity, Whan 43007, Popl s Rpblic of China ail: sxichn@63.co Rcivd Fbrary 06 Accptd for pblication 6 Jn 06 Pblishd 3 Novbr 06 Abstract Whn r stdying th vac sitch, fond that th vac diod can radiat a broadband icroav. Th vac diod is coprisd of a cathod ith a triggr dvic and planar anod, thr is not a tallic bllos avgid strctr in this dvic, so th radiation chanis of th vac diod is diffrnt fro th plasa filld icroav dvic. It is hard to copltly iitat th thory of th plasa filld icroav dvic. This papr analys th brakdon procss of th vac diod, stablishs th athatical odl of th radiating icroav fro th vac diod. Basd on th analysis of th disprsion rlation in th for of a rfractiv indx, th lctroagntic avs gnratd in th vac diod ill rsonat. Th incldd angl btn th dirction of th lctroagntic radiation and th initial otion dirction of lctron ba is 45 dgrs. Th papr isolats th lctrostatic ffct fro th baplasa intraction hn th lctroagntic radiation occrs. According to abov analyss, th disprsion rlations of radiation ar obtaind by solving th av qation. Th disprsion crvs ar also obtaind basd on th thortical disprsion rlations. Th thortical disprsion crvs ar consistnt ith th actal asrnt tifrqncy aps of th radiation. Thortical ddction and xprints indicat that th rason for icroav radiating fro th vac diod can b ll xplaind by th intraction of th lctron ba and agntid plasa. Kyords: baplasa intraction, disprsion rlation, disprsion crv, id band icroav radiation (So figrs ay appar in color only in th onlin jornal). Introdction Whn stdying th vac sitch, fond that th vac diod can radiat a broadband icroav. Th prliinary chanis analysis of th radiation [] shos that th agntic fild gnratd by th crrnt floing throgh th plasa aks th plasa bco anisotropic and inhoognos; th radiation of th plasa is a kind of slfxcitd radiation. Th disprsion rlation has not bn obtaind. * Spportd by National Natr Scinc Fondation of China (No. 0753), th Yong Scintists Fnd of Natr Scinc Fondation of China (No. 5077). Thr ar lctron ba and plasa in th vac diod, th vac diod has so siilaritis ith th plasa filld icroav dvic [, 3]. Th chanis of plasa filld dvics is ostly basd on th concpt of th Ionchannl [4]. This concpt rqirs t < 0.5 t i, t is th dration of lctron ba pls and t i is th ion plasa priod. In th xprints of th vac diod hich hav copltd, th crrnt floing throgh th vac diod is lss thn ka, th dnsity rang of th plasa is to 0 8 3, so that 0.5t iax» 750 ns. Th dration of th lctron ba that asrd is abot 00 ns, it is lss than 0.5 tiax, so that thr is th phnonon of baplasa intraction hich is siilar to th plasa filld icroav dvic in th vac /7/ $33.00

2 diod. Bt th vac diod also has iportant divrsity fro th plasa filld icroav dvic. Thr ar not th slo av strctrs of tal rippl or tal avgid in th strctr of th vac diod, and th vac diod consists of a cathod ith a triggr dvic and plat anod. Th qations in this papr ar siilar to th qations stablishd for th plasa filld icroav dvic, bt th bondary probl in this papr is vry diffrnt fro th plasa filld icroav dvic. Throgh th prliinary analysis and th contrast btn th vac diod and th plasa filld dvic, it is hard to copltly iitat th xisting thory to xplain th radiation chanis of th vac diod. Th plasa is not a passiv di, and thr ill b intraction btn th lctron ba and th plasa [5, 6]. Th intraction ill lad to varity instabilitis [7 9], and it ill ak th chanis of radiation bco or coplx [0 ]. According to th physical procss of th icroav radiation fro th vac diod, this papr stablishs a athatic odl of th procss and obtains th disprsion rlations of th odl, and thn th papr copars and analys th thortical rslts and th actal asrnt rslts of radiation.. Physical odl of icroav radiation A high spd cara is sd to tak pictrs of lctric brakdon in ordr to analy th radiation chanis. Figr (a) is th ont of triggr. Whn th diod is triggrd, th initial plasa ill b gnratd in th triggr gap, and thn lctrons of th initial plasa srfac ar drivn toards th anod by th lctric fild btn cathod and anod. Figr (b) is th ont of anodic plasa foration, th anodic plasa is gnratd by lctron bobardnt. figr (c) is th ont of cathod and anod plasa closr, plasa closr is d to th bipolar diffsion of th plasa. Whn th anod plasa has bn gnratd and th plasa closr has not occrrd, lctrons ittd fro th initial plasa ill b incidnt to th anod plasa, th vac diod ill radiat icroav in this procss. Whn th plasa closr has occrrd, th voltag drop btn cathod and anod ill b th voltag drop in th arc coln, and th radiation ill b finishd. In ordr to dscrib th physical procss, th cylindrical coordinat of ( r, q, ) is stablishd, th cntr lin of th lctron ba is th axis, and th dirction of b is th positiv dirction, so that th physical odl of th ba and th plasa is as shon in figr. It is assd that th lctron ba is a nifor ba, and th odl of lctron ba and th plasa is sytrical in th q dirction. Th radis of ba is R, th ba dnsity is n b, and ba vlocity is b bc (c is th spd of light in a vac, 0 < b <, Lornt factor is g, th plasa dnsity is n p. If b th diatr of th lctron ba is ch sallr than th radial dinsion of th plasa, th plasa can b rgardd as infinit in th radial dirction. Bcas th slfgnrating Figr. Physical iags of brakdon procss. Figr. Physical odl of ba and plasa.

3 agntic fild of th lctron ba is a poloidal agntic fild, th plasa is agntid by th agntic fild of th lctron ba, and lctroagntic avs hich propagat in th plasa along th axial dirction and th radial dirction prpndiclar to th agntic fild bco slo avs. Thr is an intrfac btn th lctron ba and th agntid plasa, th odl is dividd into to parts by th intrfac: insid of th lctron ba and otsid of th lctron ba. 3. Analysis of disprsion rlation In th discssion of th ffct of th prtrbation in th lctron ba and th plasa, it is assd that th ion is fixd, and th rlatd physical qantitis cold b xprssd as th s of th qantity ndr qilibri stat and th prtrbation qantity, as ll as A A0 + A. This papr stiplats that th sbscript 0 rprsnts th qantity ndr qilibri stat; th sbscript rprsnts th prtrbation qantity. This papr also stiplats that th sbscript rprsnts th qantity of axial dirction; th sbscript r rprsnts th qantity of th radial dirction. Th prtrbation qantity A is rlatd to xp ( jkx j t), (k and ω is av nbr and anglar frqncy rspctivly), andit st sffic th condition A0 A. AccordingtoMaxll qations, th prtrbation qantity of lctric fild st sffic th av qation. Th sybol J is th hol crrnt dnsity prtrbation of th odl. ( ) + c j J 0 ( ) For solving qation (), it is ndd to find ot th rlationship btn th prtrbation of th crrnt dnsity and th prtrbation of th lctric fild, bcas th odl is dividd into to parts: insid and otsid of th lctron ba, so th rlationship btn th crrnt prtrbation and th lctric fild prtrbation is fond ot for th to parts rspctivly, sbscript b dnots th insid of th lctron ba, hil sbscript p dnots th otsid of th lctron ba. and qation (3) is th qation of otion. nb + nb b t + ( b ) nb 0, ( ) n b + + b ( b ) vb t n b( + b ) ( 3 ) According to th condition of anglar sytry 0, th q prtrbation dnsity insid of th lctron ba is obtaind by solving qation (). nb0 ( r br) nb j + kb k r r b0 Bcas th agntic fild insid of th lctron 0nb0b0r ba Bb0 is obtaind by th Apr dr circital thor, and th ti drivativ of r is dt + b0 t r br, th agntic fildcanbrittnas 0nb0b0 Bb0 br, and thn b Bb0 is infinitsial, it can b ignord. For th prtrbation insid of th j( kb0) lctron ba, th linarid otion qation is b + ( b0 ) b ( t g b b0 B b). ( 4 ) Bcas of b jbb and th condition of anglar sytry, th prtrbation vlocity is obtaind. j b b g( kb0) ( 5) j j b0 b br br g kb0 r Sbstitt b, br and n b into Jb ( nb0b + n bb0), and thn th prtrbation of crrnt dnsity is obtaind ith n b0 b. 0 J J br b j 0 b jb0 b br g k b0 r j 0 b j k g k r r r r r ( ) r b0 b0 ( rbr) b0 b b0 + b ( 6) 3.. Th prtrbation insid of th lctron ba Th initial vlocity of th lctron ba is in th dirction, th ba initial dnsity is n b0, and ba initial vlocity is b0. Th lctron ba st sffic th continity qation and th qation of otion, qation () is th continity qation 3.. Th prtrbation otsid of th lctron ba Th initial vlocity of th plasa is ro, th initial plasa dnsity is n p0. Th plasa st also sffic th continity qation and th qation of otion, qation (7) is th continity qation and qation (8) is th qation of 3

4 otion. np + np p t + ( p ) np 0 ( 7) n p p + ( p t ) p + np( + p Bp) 0 ( 8) By th sa thod insid of th lctron ba, th prtrbation dnsity and th prtrbation vlocity otsid of th lctron ba ar obtaind ith B p0 c. np0 ( r pr) np j + kp ( 9) r r j + c pr r p c p c j c p pr c p c ( 0) Th av qation of qation () can b rittn as k ( k ) k + ( I + ) 0 c by Forir transfor. Stiplat T k k k ij i j d i j + ( d j + j c i i) ith k r k sin y, k k cos y, k k + kr (y is th incldd angl i j btn b0 and k ), and d ij, so that th av 0 i ¹ j qation can b rittn as T 0 finally. If th nontrivial soltions xist for th av qation, dtrinant of th cofficints of th qation ill b dt ( T ) dt k k k i j d i j + ( d j + j 0. 3 c i i) ( ) Sbstitt vry coponnt of ε into qation (3), and thn th rfractiv indx of lctroagntic avs gnratd in th vac diod is obtaind. n kc + + rr + rr r r + cosy + siny + cos y sin y + cos y sin y rr r r ( 4) Sbstitt pr and p into Jp n p0p, and thn th prtrbation of crrnt dnsity is obtaind ith n p. 0 c Jpr j 0 p j pr 0 p p c c ( ) c Jp j 0 p r + j p 0 p p 3.3. Th analysis of th rfractiv indx c c p0 Th rfractiv indx is analyd in th ( r, o, ) plan, and r is rplacd ith j kr. Th hol crrnt dnsity prtrbation on th intrfac of baplasa is J Jb + J p, and bcas of j 0 J, c so that rr r, r hr b p rr c c i b kr b0 r + p k b0 c, c i b krb0 r p k b0 c + k b r b0 p kb0 ( ) ( ) c Whn n, th rsonanc of lctroagntic avs ill occr. If n, th dnoinator of qation (4) st b ro. Sbstitt th coponnts of ε into th dnoinator, and thn th folloing xprssion is obtaind. F(, k) + cosy + sin rr y + cos y sin y + cos y sin y r b ( k sin y cos y r b0( )) ( k ) p c Stiplat F(, k) 0, and thn b ( k sin y cos y r b0( )) ( k ) b0 r b0 p c 0 ( 5) and th straing instability disprsion rlation of lctron ba and plasa is b p 0. ( 6) ( k ) b0 It is asy to find that th conforation of qations (5) and (6) is vry siilar. Thr is only on tr on th right of th qal sign of both to xprssions, this tr is ro; thr ar thr trs on th lft of th qal sign, th first tr of ach xprssion is, th scond tr is rlatd to th lctron ba, th third tr is rlatd to th plasa. For th third tr on th lft of th qal sign, th qation (5) taks accont of th lctron cyclotron casd by th agntic fild 4

5 of th lctron ba, and qation (6) dos not tak accont of th lctron cyclotron. According to th phnonon obsrvd in th xprint, > p and p» b, and a c c 0 br nb. Whn th vlocity of th b 4 lctron ba is b0 0.6 c, th radis of ba is c and th crrnt of th ba is ka, and thn a is approxiatly qal to 0., and it is consqntly assd that c, as ll as c». In fact, th divrsitis of qations (5) and (6) ar ainly cntralid in th scond tr, on th lft of th qal sign. Th xprssion of F(, k) 0 can b transford into qation (6) in to sitations. Th first sitation: k r 0. Th scond sitation: sin y cos y 0. If kr 0, according to th av qation k ( k ) k + ( I + ) 0, c th dirctions of k and ill b in th sa dirction, th vac diod cold not radiat lctroagntic avs, and this infrnc dos not atch th phnonon obsrvd in xprint. To s p, th prcondition of F(, k) 0 is sin y cos y 0, as ll as y 45. All of th abov analyss indicat that if th lctroagntic avs gnratd in th vac diod ar xtraordinary avs, th avs ill rsonat. Th incldd angl btn th dirction of th lctroagntic radiation and th initial otion dirction of lctron ba is 45 dgrs. Whn th lctroagntic radiation occrs, th lctrostatic ffct can b isolatd fro th baplasa intraction; thrfor, th lctroagntic radiation in th dirction of y 45 can b discssd sparatly Th disprsion rlation of th lctroagntic radiation Th disprsion rlation of th lctroagntic radiation gnratd in th vac diod is obtaind by solving th av qation in th ( r, q, ) cylindrical coordinat. Rsolv th av qation () into th av qations of axial dirction and radial dirction. r r r + r c j J 7 0 ( ) r + r c r j J r 8 0 ( ) r For th soltion insid of th lctron ba: sbstitt J br into qation (8) and thn find ot th rlationship of th lctric fild prtrbations in dirction and r dirction. br b b0 k gc kb0 j r b k c g b ( 9) r r r b p b 0, r hr p b k c g This qation is th odifid Bssl s diffrntial qation of ro, and b is finit in th cntr of th ba scilict b <+. Th soltion of th partial diffrntial qation can b rittn as b ai0 ( pr) 0 < r R. ( 0) For th soltion insid of th lctron ba: according to th radiation in th dirction of rsonanc, copar th ral parts and th iaginary parts of th prtrbation vlocity otsid of th lctron ba. c R pr [ p] c I[ p] p c c arctan y c R r p [ p ] c I[ pr] r p c c tan y Bcas of y 45 scilict tan y arctan y, and bcas th postlat c is sfficd, so that th iaginary parts ar ch gratr than th ral parts, th prtrbation vlocity otsid of th lctron ba only inclds th ral parts, and th xprssion is shon in (), th prtrbation of crrnt dnsity is rittn as (). pr p j j pr c p c Jpr j 0 p c Jp j 0 p pr p c ( ) ( ) According to th thod insid of th ba, on can find ot th rlationship of th lctric fild prtrbations hich ar otsid of th ba. pr k j p k c c r p ( 3) Thn sbstitt br into qation (7), th partial diffrntial qation of b is obtaind. Th partial diffrntial qation of p sa thod. is obtaind by th 5

6 r r r p q p 0, r hr q p k c This qation is also th odifid Bssl s diffrntial qation of ordr ro, and th soltion of th qation can b rittn as: a I ( qr) + a K ( qr) r R. ( 4) p Th sybol I 0 is th odifid Bssl fnction of th first kind of ordr ro, and th sybol K 0 is th odifid Bssl fnction of th scond kind of ordr ro. c Bcas of B t, th rlationship btn th lctric fild prtrbation and th agntic fild prtrbation can b obtaind. k b b0 b + j gc k c g b0 b Bbq ( 6) r b k c g B pq j k p c c p c c r p Finally, th thr bondary conditions can b rittn as ( 7) k k b b p p b p r: g3 k b0 p r q r g k b0 b p ( ) ( ) b p g k b0 b c p q: k k 0 p r q r : 0 ( 8) b p c c On th intrfac btn th ba and th plasa scilict r R, th prtrbations st sffic th bondary conditions in thr dirctions of th ( r, q, ) cylindrical coordinat. In th r dirction, Gass thor is sd to obtain th bondary condition òd ds òdrdv òn d V, and th prtrbation dnsity n is th s of th prtrbation Sbstitt th soltions of lctric fild b and p into qation (8). If th qation st of th lctric fild has nontrivial soltions, th dtrinant of th cofficints of th qation st ill b ro. By th coplx athatical oprations, can obtain th disprsion rlation of th lctroagntic radiation. I0 ( pr) I ( pr) k p b p b + g k 3 b0 c g ( kb0) p p b + g( kb0) c c p c ( 9) dnsity insid and otsid of th ba, as ll as n nb + n p. In th q dirction, th agntic fild is continos, and thn Bb q B p q. In th dirction, th lctric fild is continos, and thn b p. Sbstitt n b and n p into th Gass thor, and thn nb0 br pr j b 0( kb0) nb0 ( k ) 0 b0 r b + j n p0 0 np0 0 pr p ( 5) 4. Copar th disprsion crvs to th radiation spctrs R is th radis of th lctron ba, hich is xprssd as a ltipl of th avlngth, and thn R l. Sti k kb0 n plat W, Z, a c p p, c, n b b b b b 6

7 Figr 3. Disprsion crvs. Figr 4 Th tifrqncy ap of 700 A ba crrnt. p b so that ( W ), th disprsion rlation (9) k Z is noralid by b, and qation (30) is obtaind. In th lab, th actal asrnt radiation has bn obtaind ndr th condition of 700 A lctron ba crrnt, acclrating voltag 80 kv, and lctron ba radis c, hich is shon in figr 4. Th actal asrnt tifrqncy ap shon in figr 4 is obtaind by Garbor transforation. According to ths data of xprint, it can obtain that (a) Th ratio of th lctron ba vlocity to th light spd b» 0.8 (b) Th ratio of th ba lctron frqncy to th lctron cyclotron frqncy a 0. (c) Th ba lctron frqncy b rad/ s, and corrsponding frqncy of radiation f.6 GH So that 0.9, a 0. and b 0.8 ar slctd to plot th disprsion crv, hich is shon in figr 5. Figr 5(a) is th hol tndncy of th disprsion crv, and figr 5(b) is th lo frqncy dtail. Fro th disprsion crv of figr 5, th to bands of radiation fro th diod ar 0.65fb.fb and 0 0.8f b in thory, and anothr for of xprssion, GH and GH. Fro th tifrqncy ap of figr 4, th to frqncy bands in th xprint ar GH 4 GH and GH, thos to frqncy bands ar incldd in th to frqncy bands of th disprsion crv shon in figr 5. Th radiation of lor ba crrnt and lctron vlocity is also asrd in th lab. If th lctron ba crrnt is 0 A, th acclrating voltag is 80 kv, and th lctron ba radis 0.5 c, thn it ill b asy to calclat that b» 0.5, 0., a 0.0 and fb 0.9 GH. So that 0., b 0.5, a 0.0 is slctd to plot th disprsion crv, hich is shon in figrs 6 and 7 is th tifrqncy ap. It is convnint to find that th frqncy of radiation fro th diod is abot f b fro figr 6, and th actal asrnt radiation is also on th sa frqncy. Whn th ba crrnt and th lctron vlocity ar rathr lo, th disprsion rlation of th thortical ddction b I0 I b ( W ) Z b ( W ) Z g c g3 c + W W Z W a ( W Z) W a b c c g ( W ) + Z W a W a ( W Z) ( 30) In th xprints, th diatr of th ba is liitd to th cntitr lvl, and th crrnt of ba is lss than ka, a < 0.5 can b stiatd. Th avlngth of th radiation is also liitd to th cntitr lvl, so that. It is assd that c g according to rfrnc [4]; hil th critical val of c g cold b slctd to plot th disprsion crvs. Finally, 0.95 and a 0. ar slctd to plot th disprsion crvs for diffrnt initial vlocitis of th lctron ba, hich is shon in figr 3. approxiats a horiontal lin, and th tifrqncy ap also approxiats a horiontal lin. Fro th crvs hich hav alrady bn plottd in this papr, thr is not an intrsction point of th crvs of th disprsion rlation and th straight lin of light vlocity, and th frqncy of th lctroagntic avs that radiat fro th diod is gratr than g b, in othr ords, th frqncy of radiation xcds th frqncy of th plasa p. Th ndlation of th disprsion crvs bcos sallr and sallr ith th dcras of lctron ba vlocity. 7

8 Figr 5. Disprsion crv of 700 A ba crrnt. Figr 7. Th tifrqncy ap of 0 A ba crrnt. Figr 6. Disprsion crv of 0 A ba crrnt. 5. Conclsion () This papr drivs th disprsion rlation of th icroav radiation gnratd by th brakdon procss of th vac diod, and th disprsion crvs ar plottd by th disprsion rlation. Th thortical ddction and th xprints indicat that th icroav radiation gnratd in th vac diod is th consqnc of intraction btn th lctron ba and th agntid plasa. () Th lctroagntic avs gnratd by th baplasa intraction in th vac diod ar xtraordinary avs, th avs ill rsonat ith th agntid plasa, and th incldd angl btn th dirction of th lctroagntic radiation and th initial otion dirction of lctron ba is 45 dgrs. Whn th lctroagntic radiation occrs, th lctrostatic ffct can b isolatd fro th baplasa intraction. Thrfor th lctroagntic radiation in th dirction of y 45 can b discssd sparatly. (3) Th thortical disprsion crvs drivd in this papr ar consistnt ith th tifrqncy ap. Th ndlation of th disprsion crvs bcos sallr and sallr ith th dcras of th lctron ba crrnt. In th nd, th disprsion crv ill b siilar to a horiontal lin. Th probl discssd in this papr is only for th sitation of lo lctron ba crrnt, and th probl of highr lctron ba crrnt is ndd to discss this frthr. Rfrncs [] Chn S X t al 008 High Por Lasr and Particl Bas (in Chins) [] Gobl D M t al 996 PASOTRON HighPor Microav Sorc Prforanc. (Dnvr, CO, Unitd Stats: Intns Microav Plss IV) [3] LiPKt al 997 Acta Phys. Sin (in Chins) 8

9 [4] Whitt D H, Ssslr A M and Dason J M 990 Phys. Rv. Ltt [5] Karbshv N I and Rostoyan V 008 Phys. Ltt. A [6] Brt A, Firpo M C and Dtsch C 004 Phys. Rv [7] Watson K M, Bldan S A and Rosnblth M N 960 Phys. Flids 3 74 [8] Bldan S A, Watson K M and Rosnblth M N 960 Phys. Flids [9] Brt A, Dickann M and Grillt L 00 Ann. Gophys. 8 7 [0] LiSGt al 000 I Trans. Plasa Sci [] S D and Tang C J 009 Phys. Plasas [] S D and Tang C J 0 Phys. Plasas