RF loss in and leakage through thin metal film

Transcription

1 SLAC-PUB-794 August 1998 RF loss in and leakage though thin metal film Xintian Eddie Lin Submitted to IEEE Tansactions on Micowave Theoy and Techniues Stanfod Linea Acceleato Cente, Stanfod Univesity, Stanfod, CA 949 Wok suppoted by Depatment of Enegy contact DE AC 76SF515.

2 RF loss in and leakage though thin metal lm Xintian Eddie Lin Abstact Taditional RF enclosues ae fomed by metal bulk mateial, in which the eld decays exponentially ove a skin depth p = =!. With typical bulk dimension at least one ode of magnitude highe than, the eld outside the metal enclosue is then negligible. The suface esistance R s pesented to the RF eld is 1=. This aticle pesents analysis of an RF shield fomed by thin metal lm. We nd that a metal thickness of the ode of skin depth, can povide an excellent RF shielding due to eection at the inteface. The suface RF loss can be educed with pope choice of metal thickness, and heating can be educed. The thin metal lm also facilitates cooling on the back of the lm via themal conduction. Fo a vey thin coating, the suface RF loss is invesely popotional to thickness, while inductance is popotional to thickness, and thus it is possible to customize the acceleato beam impedance though caeful choice of coating thickness. Keywods Thin metal lm, RF leakage, impedance. I. Intoduction METAL coatings ae widely used in many applications, one of which is RF shielding. Fo example, ceamic vacuum chambes ae used in acceleato beam monitos, injection kicke and feedback systems, and metal coating is applied to the inne ceamic wall to povide a conduction path and pevent static chage accumulation. Theefoe, it is impotant to chaacteize the impedance and uantify the RF shielding. Piwinski[1] has analyzed the RF impedance and penetation of metal coating on ceamics though eld matching, but did not obtain the minimum suface esistance at 1. Chao[] has deived the thin lm impedance by assuming shot cicuit bounday condition, which does not apply in this cicumstance. Jackson[] found the impedance of two thin metal layes assuming no RF leakage. Couant and Month[4] also gave the suface impedance of metal lm on metal. In this aticle we analyze the thin lm suface impedance and RF leakage assuming outgoing wave bounday condition and also povide the effective impedance pesented to a chaged paticle beam. II. Suface impedance To simplify the calculation, we will use the petubation appoach instead of a self-consistent eld matching solution. The elds ae obtained by st assuming a pefectly conducting bounday, and then applying a small suface impedance as a petubation. The dieence between the petubation appoach and full eld matching is of the ode of 1 (R s= )(!=c)b; thus fo a cm adius coppe pipe, at 4.77 GHz, the coection is This wok is suppoted by Depatment of enegy unde Contact No. DE-AC-76SF515. Xintian E. Lin is with the Stanfod Linea Acceleato Cente, Stanfod Univesity, CA. eddie@slac.stanfod.edu A. Thick metal Thee ae many textbook analyses of RF elds inside metal; howeve, fo convenience and compaison, a bief discussion is povided hee. In the case of a thick metal coating, the elds decay exponentially away fom the suface[5]: H y = H e (1 i)x= (1) E z = s H e (1 i)x= ; () whee suface impedance s = (1 i) p! =, and x is the nomal coodinate. The implicit dependence of e ikz z i!t on all uantities is suppessed fo bevity. The electic cuent J z follows fom J z = E z = s H e (1 i)x= : () It is evident fom E. that the magnitude of elds decay exponentially with penetation into the conducto, and that has the signicance of the depth whee elds dop to 1=e of thei suface value. Also notice that the phase lags by x= adian at depth x into the conducto. Theefoe, efeenced to the suface, the aveaged cuent ow <J z >=J e x= cos x eveses the diection at depth x = 1 as shown in Fig. 1. cuent Fig. 1. B. Thin coating x in unit of skin depth Cuent distibution in metal. Without loss of geneality, we assume a cylindical geomety shown in Fig., whee egions 1 and ae vacuum and metal espectively. Region can be a dielectic o (4)

3 othe metal and extends to innity. We will label uantities with a subscipt coesponding to each egion unless dened othewise. holds. We can futhe simplify E. 11 to 1 = c = i tan 1 k (1) if which educes to j( = c ) sin k j jcos k j; (14) 1 1 z p 1 (15) Fig.. Metal coating in a cylindical chambe with inne adius 1. The coating thickness is 1. When the metal coating is thin compaed with skin depth, the solution of H and E have two tems, H = ae (1 i)x= + be (1 i)x= (5) E z = c (ae (1 i)x= be (1 i)x= ) (6) whee a and b epesent the outgoing wave and eected wave amplitude espectively. Hee c =(1 i) p! = epesents the tansmission line chaacteistic impedance of the metal, and x = 1. The impedance dened at the inteface = can be expessed as = c = k (1) ih (k )! H (1) (k ) ; (7) a esult of outgoing adial wave bounday condition in medium, whee H (1) is the Hankel function of the st kind. The adial popagation constant k satises k = (! c ) k ; (8) z whee is the elative pemittivity. When k 1, the expession fo c educes to that of a plane wave = c = k! = k (!=c) (9) Fo a dielectic the chaacteistic impedance c is oughly ( p 1= ) assuming k z =!=c. It is of the ode = 77. Meanwhile c fo coppe at 11 GHz is :8(1 + i). If the mateial in egion is metal, eplace by (1 + i =! ). Unde the condition! =, we ecove the expession! c =(1 i) : (1) Fom E. 5 and 6, one can elate the suface impedance 1 at = 1 with by impedance tansfomation 1 = ( = c ) cos k i sin k c cos k i( = c ) sin k ; (11) whee k =(1+i) p! =. In the case that mateial is a good conducto, and mateial is a poo conducto, dielectic o even ai with =1:564[6], the condition j = c j1 (1) fo dielectic o (16) fo metal in egion. With =5:5 and =5:8e7(m) 1 fo coppe lm, E. 15 euies 1: _A. In the case of coppe lm on stainless steel with =:, E. 16 euies :14. E. 1 can be undestood physically fom the open cicuit bounday condition at = as E. 1 suggests. Chao[] has deived the thin metal lm impedance unde the closed cicuit bounday condition at =, whichdoes not apply in this cicumstance. In a numeical example of =5:5 and k z =:77!=c, the value of 1 fom E. 11 is plotted in Fig.. The eal 1 nomalized to Rs Metal on metal, σ /σ =. Metal on dielectic coating thickness in unit of skin depth Fig.. Suface esistance of a thin metal lm as a function of lm thickness. The solid lines epesent <(1)=R s, and the dashed lines epesent =(1)=R s. The coating is chosen to be coppe and substate is dielectic with = 5:5 o stainless steel. pat of 1, plotted as solid line, exhibits a minimum about :9R s at = 1:57 1. And as expected, when goes to innity, it appoaches R s. A simila behavio fo coppe lm on stainless steel ae shown in the plot too. Recall that in thick conducto, thee ae cuent owing in the opposite diection at depth geate than 1 due to the phase lag. The eduction of the suface loss of a thin lm about 1 thick is a diect esult of eliminating invese cuent ow in the conducto. Theefoe, the aveage cuent density nea the conducto suface is educed. Fig. 4 illustates the distibution of amplitude and phase

4 1 1 abs(jz) x in unit of skin depth 1 8 nomalized to Rs 1 5 angle(jz) x in unit of skin depth coating thickness in unit of skin depth Fig. 4. The amplitude and phase of the electic eld in the conducto. The metal lm thickness is 1. Fig. 5. Radiation impedance R of a thin metal lm as a function of lm thickness. Substate is dielectic with = 5:5. of cuent in the metal lm with = 1. The amplitude is nomalized to the suface value of an innite thick metal. Notice that the phase lagging is within 9 o, so thee is no evese cuent. And the electic eld on the thin lm suface is smalle than that of the innite thick metal. III. Radiation Impedance The RF leakage though the metal lm, chaacteized by the powe adiated to media pe unit aea, can be expessed as P = 1 Re(E z H )j = = 1 Re(jE zj )j = : (17) If we dene a tansfe impedance t to elate the electic eld E z at = to the suface magnetic eld H at = 1 as t E zj = = H j =1 cos k i( = c ) sin k ; (18) then the powe leakage becomes P = 1 Re(j tj )jh j j =1 1 R jh j j =1 ; (19) whee we dene adiation impedance R so we can compae to suface impedance 1, whose eal pat indicates total powe loss. In the case that jk j > 1, we can futhe simplify R = Re( j tj )=R s 8R s j c j e = : () The numeical value of R is plotted in Fig. 5. The exponential dependence of R on the lage lm thickness is obvious. But even with a lm 1:5 skin depth thick, adiation into media only amounts to 1 4 that of the total powe loss. This is a esult of nea total eection at the inteface between media and due to the lage diffeence between the chaacteistic impedance c and c. The eection coecient fom media to, c = c (1) c + c is nealy one. Thus the factional powe tansmitted though the inteface is 1 j j = 4R s j c j : () If we include the attenuation in the metal lm we would have R = 4R s j c j e = : () R s The dieence between E. and is a esult of the non unitaity of the S-matix when a mode with complex impedance is pesent[7]. IV. Multi-laye Coatings The esults in the pevious sections can be genealized to multi-laye coatings though epeated use of E. 11 and 18. A n-laye coatings system is dened by a one-dimensional aay j, j fom 1 to n 1, which species the inteface coodinate between medium j and j +1. Wehave j whee 1 E z H j =j 1 = jc ( j = jc) cos j i sin j cos j i( j = jc ) sin j (4) tj E zj =j+1 j+1 = ;(5) H j =j cos j i( j = jc ) sin j j = k j ( j+1 j ) (6) jc = k j = k j (7)! j (1 + i j =! j )! j j (1 + i j =! j ) k z (8)

5 and outgoing wave bounday condition n 1 = nc : (9) Then we obtain the tansfe impedance fom = 1 to = n 1 t = Q n j=1 tj Q n j= j = and the adiation impedance t(n 1) Q n j= [cos j i( j = jc ) sin j ] () eal and imaginay pat of thin coating: 1 skin depth thin coating:.5 skin depth R = < j tj nc : (1) Taking the PEP-II B-factoy[8] as an example, the IR chambe inside the detecto is a 4 cm long double-wall beyllium pipe. The walls ae.8 mm and.4 mm thick espectively with 1 mm gap fo cooling wate. The ID of the wall is 5 cm. Even though the beyllium walls epesent less than 5 skin depths at 16 khz evolution feuency, the adiated powe into the detecto by a A beam with 5% gap is only watts. V. Beam Impedance In addition to conning RF powe, metal enclosues ae also used to tanspot chaged beams. An impotant gue of meit is the beam impedance dened as b (!) = R L E z(!; z)e i!z=v dz ; () I b (!) whee v is the speed of the chaged beam. Fo a long beam pipe, E z is popotional to e i!z=v,thus b = E zl I b = 1H 1 I b L = 1 L 1 ; () whee we have used the fact that E z is unifom inside egion 1 fo highly elativistic beam and also H 1 = I b = 1. Because b is the same as the suface impedance 1 aside fom a geometic facto, we will continue using 1 when we efe to beam impedance. A. Impedance spectum A bunched beam with length b has feuency contents up to! b = c= b, and this sets a natual feuency efeence. A efeence fo metal thickness is povided by the skin depth b = (4)! b at! b. With a beam of b = 1 cm, the feuency ange! b = =4:77 GHz and b =:96 m in coppe. Beam impedance is eadily available fom E. 11, and plotted in Fig. 6. Wehavechosen =5:5, coppe lm and b = 1 cm. The metal lm thickness is specied in unit of the skin depth b. The thick coating limit exhibits a nomal suae oot dependence on feuency. When the coating becomes thinne, the elds inside metal become gadually. thick coating feuency x 1 9 Fig. 6. The beam impedance as a function of feuency. The solid and dashed lines epesent eal and imaginay pat of the impedance espectively. The bunch length b is chosen to be 1 cm. moe unifom. The esistance, i.e., the eal pat of the impedance, theefoe appoaches the DC limit, 1=. As a esult, the esistance becomes less feuency dependent. This scaling eventually beaks down when E. 15 does not hold. Then the metal lm becomes too thin to shield the RF eld, i.e., when it can not conduct all the image cuent. At the limit of zeo metal thickness, the esistance of metal goes to zeo. While esistance is inceased acoss most of the spectum with a thinne coating, the eactance, i.e., the imaginay pat of the impedance, pedominately inductive, is educed as a esult of less volume in the metal to stoe magnetic enegy. The esults can be undestood ualitatively by expanding E. 1 to obtain 1 = 1 [ ( ) 4 ] whee we have used the Taylo expansion i! ; (5) x tan x =1 1 1 x 45 x4 + O(x 6 ): (6) Just as we obseved in Fig. 6, the esistance is invesely popotional to at DC and has a uadatic incease with feuency because / 1= p!, while inductance is popotional to. If mateial is anothe metal, E. 11 educes to 1 = 1 cos x i sin x x; (7) sin x + i cos x whee = p = and x =(1+i)=. Fig. 7 illustates the beam spectum of a1cm bunch, with = =:. At low feuency, the RF elds penetate though metal, and the impedance follows that of metal. When penetation depth becomes compaable to the coating thickness at high feuency, the impedance appoaches that of metal

6 eal and imaginay pat of thick coating thin coating: 1 skin depth thin coating:.5 skin depth By substituting E. 1 o 7 in E. 41 fo thin metal coating, we have k thin = L c ( ) 4 1 b b (44) with = <[ 1+i ( 4 ) 1 fo metal coating on dielectic and = <[ 1+i ( 4 ) 1 1=4 e y y tan x dy] (45) cos x i sin x sin x + i cos x y 1=4 e y dy] (46) feuency x 1 9 Fig. 7. The beam impedance as a function of feuency. The solid and dashed lines epesent eal and imaginay pat of the impedance espectively. The bunch length b is chosen to be 1 cm.. In the tansition feuency ange, the impedance has a less feuency dependent esistance and inductance. The smalle the, the faste the initial ise, and a close esemblance to Fig. 6. B. Eective Impedance One way to uantify the eect of impedance on a beam is the beam spectum weighted impedance aveage. Thus we dene uantity k = 1 <( b (!))ji(!)j d! ; (8) which gives enegy loss. We do not need to take the eal pat when integating fom 1 to 1 because =( b )isan odd function of!. Fo a Gaussian bunch pole (s) = the cuent spectum is also Gaussian p b e s = b ; (9) I(!) =e (!=c) b = : (4) Substituting E. and 4 into 8, we obtain k = L 1 1 <( 1 ) e (!=c) b d! : (41) If the thick metal 1 is used, we obtain the familia esult k thick = L c ( ) 4 1 b b ; (4) whee we applied the denition of Gamma function (z) = 1 y z 1 e y dy: (4) fo metal lm on metal. We have dened = p = and x =(1+i)(= b )y 1=4. Anothe gue of meit that entes diectly into the beam instability is the aveaged b =![9] whee and ( b! ) ef f = P 1 p= 1 b(! )=! h l (! ) P 1 p= 1 h l(! ; (47) ) h l (!) =(! b c )l e (!=c) b ; (48)! = p! + l! s : (49) The uantities! and! s ae evolution and synchoton feuencies espectively. Fo boad band impedance, we can eplace the sum by anintegal, and the eal pat vanishes because Re( b )isaneven function of!. The vanishing esult holds tue in the limit! s!. Substituting the metal suface impedance into E. 47, we obtain ( b! ) ef f = L b i 1! b (l + 1) 4 (l + 1) : (5) Similaly, the eective impedance of thin metal coating is ( b! ) L b (l + 1 ef f = i ) 4 1! b (l + 1) (51)! ;l with 4 e y = =[ (1 + i) 1 y l! ;l (l + 1 ) 4 tan x fo metal on dielectic and dy] (5) = =[ (1 i) 1 cos x i sin x! ;l (l + 1) 4 cos x i sin x yl 4 e y dy] (5) fo metal on metal. The value of 's ae plotted in Fig. 8. In ode to t the lines in the same ange, we plot 1= and ;1. The value! of geneally scales as 1= and! ;1 is popotional to at small, which is the esult of E. 5. The! ;1 of metal lm on metal inceases at smalle value because of the eld penetation. Fo metal lm on dielectic,

7 effective impedance metal on metal, σ /σ =. metal on dielectic metal on metal, σ /σ =. [4] E. D. Couant and M. Month, \Tansvese esistive wall instability of an o-axis ibbon beam in a cicula chambe", BNL Repot BNL-5875, Unpublished [5] J. D. Jackson, Classical Electodynamics. nd ed., New Wok: Wiley, 1975, pp [6] D. R. Lide, Handbook of chemisty and physics. 77th ed., CRC pess, 1997, pp. 6-. [7] Xintian E. Lin, Tansvese Wakeeld of Waveguide Damped Stuctues and Beam Dynamics, Ph.D. thesis, SLAC-R-47, pp [8] PEP-II An Asymmetic B-factoy, Conceptual Design Repot, SLAC-418, 199. Unpublished [9] A. W. Chao, Physics of Collective Beam Instabilities in High Enegy Acceleatos. New Yok: Wiley, 199. pp.. [1] S. Heifets et al. \Impedance study fo the PEP-II B-factoy" SLAC/AP-99. Unpublished coating thickness in unit of skin depth Fig. 8. The eective impedance plotted as a function of coating thickness. The solid lines ae 1= and dashed lines ae ;1.! and ;1 can be taded against each othe with thei! poduct oughly constant. Fo metal lm on metal, the minimum! ;1 is :61 and :4 fo = =: and : espectively. The coesponding enegy loss is and. times moe. Since dipole impedance is also popotional to suface impedance 1, it follows the same analysis as longitudinal impedance. Xintian E. Lin was bon in Chengdu, China, in He eceived the B.Sc degee in physics fom the Univesity of Science and Technology of China in He subseuently attended Univesity of Califonia, San Diego, and eceived the Ph.D. in physics in Fom 1995 to 1997, he woked as eseach physicist on the SLAC B-factoy RF design. Since 1997, he is with Acceleato Reseach Depatment. His eseach inteest is in diamond coating and design of high gadient acceleato stuctues, wakeeld of waveguide damped cavities and beam dynamics. VI. Conclusions We have deived the RF and beam impedance of a thin metal coating. The suface impedance can be educed by 8% if metal thickness of about 1 is chosen; this eduction esults fom eliminating evese cuent ow. The RF leakage though thin metal lm is vey small due to the lage mismatch of impedance. It is then possible to attach a high themal conductivity mateial, like diamond, to the back of the metal lm to facilitate cooling. The beam impedance of thin metal lm is also deived. Fo a thin coating, it is found to have a athe smooth esistance spectum. The educed inductance at smalle coating thickness may be useful in educing bunch lengthen in high cuent stoage ing whee esistive wall contibution is elatively lage[1]. A coppe coating of :6 b thick on stainless steel educes the inductance by 9% compaed with thick coppe wall. Acknowledgments The authos would like to acknowledge helpful discussions with D. Noman Koll. Refeences [1] A. Piwinski, \Penetation of the eld of a bunched beam though a ceamic vacuum chambe with metallic coating", IEEE tansactions on Nuclea Science, Vol 4, No., June 1977, pp [] A. W. Chao, Physics of Collective Beam Instabilities in High Enegy Acceleatos. New Yok: Wiley, 199. pp. 71. [] J. D. Jackson, SSC-N Unpublished.