Transverse Wakefield in a Dielectric Tube with Frequency Dependent Dielectric Constant

Transcription

1 ARDB-378 Bob Siemann & Alex Chao /4/5 Page of 8 Tansvese Wakefield in a Dielectic Tube with Fequency Dependent Dielectic Constant This note is a continuation of ARDB-368 that is now extended to the tansvese wakefield. The basic efeence emains Geoges Dôme s. The tansvese wakefield poduced by a unit chage paticle located at with espect to the symmety axis of the dielectic tube is () cz H w s υ () s V = λ R C m (.) whee exp( iu) υ() s = Re d π. (.) ( ) ( ) + H H ( ) Definitions ae the same as in ARDB-368 ω s k = ; = kr; u =, (.3) c R and the sign convention fo s is the same namely s > fo tailing paticles, and the integation must be pefomed teating as a complex vaiable with a small negative imaginay pat. Eq. (.) can be evaluated using contou integals and the Residue Theoem when appoximate expessions fo the Hankel functions ae used. Thee is a cut along the negative eal axis when the exact expessions ae used, and the contou integal appoach loses its appeal. Instead the symmety of the agument below the eal axis can be used to ewite eq. (.) as exp( iu) υ() s = Re d π. (.4) ( ) ( ) + H H ( ) Appendix A pesents esults fo fequency independent including appoximate expessions fo the wakefield. Fequency Dependent Dielectic Constant As in ARDB-368 conside a mateial with only one esonance at fequency ω α α = = (.) ω ω + iωγ ω + iγ whee = kr= ωr c, = ωr c, and γ is nomalied to in the ight hand expession. Assume many of the esults fom Appendix B of ARDB-368: ) Eq. (.4) emains valid with a fequency dependent. ) Poles that ae intoduced move into the uppe half plane fo γ >. The esults ae shown in Figue fo vaious values of. It is seen that thee is almost a facto of two diffeence between = and = 5 at small distances.

2 ARDB-378 Bob Siemann & Alex Chao /4/5 Page of Tansvese Wakefield fo a Single Resonance = = 5 = 5 = = 5 υ (u) u = s/r Tansvese Wakefield fo a Single Resonance = = 5 = 5 = = 5. υ (u) u = s/r Figue : Tansvese wakefield fo diffeent values of, γ =..

3 ARDB-378 Bob Siemann & Alex Chao /4/5 Page 3 of 8 Fequency Specta Fequency Specta fo Diffeent = ω R/c (γ =.) = 5 = 5 = = = ωr/c Figue : Fequency Specta fo diffeent, α =. Fequency specta ae plotted in Figue. The small diffeences in the high fequency specta lead to the behavio at small u. Tansvese Impedance Connecting υ () s to the tansvese impedance pe unit length Z ( ω) L, In tems of the quantity intoduced in eq. (3.4) we have Z iωs c ( ω) 3 R υ( s> ) =i dωe cz (.) L ( ) B = ) ) ( ) + H H ( ω) Z Z ωr =i B L π R c whee ω is consideed to have an infinitesimal negative imaginay pat. Afte the impedance is obtained, it should have thee popeties: (.3) (.4)

4 ARDB-378 Bob Siemann & Alex Chao /4/5 Page 4 of 8 ( ) When ω is confined to the eal axis, Re Z ( ω) Im( Z ( ω) ) When ω is confined to the eal axis, Re Z ( ω) should be an odd function of ω, and is an even function of ω. This assues the wakefield is eal. ( ) should be positive fo positive ω and negative fo negative ω. This assues enegy consevation. Z ω should have no singulaities in the lowe half complex ω-plane. This assues causality. When ω is extended to cove the entie complex ω-plane, ( )

5 ARDB-378 Bob Siemann & Alex Chao /4/5 Page 5 of 8 Appendix A: Fequency Independent Dielectic Constant Dôme gives the tansvese wake fo diffeent aimuthal mode numbe, m, elativistic and non-elativistic velocities and fo elative pemeability, µ > in eqs. () and (5). The dependence on the dielectic constant that is outside the integal in eq. () should be bought into the integal to allow fo a vaiable dielectic constant. These equations have been simplified by taking m =, µ =, and β =. Using expessions fo A, B, Z H, and the deivative of the Hankel function gives eq. (.) () Z H w s () s λ R υ =. (3.) whee exp( iu) υ() s = d π. (3.) ( ) H ( + ) H ( ) The physical esult is the eal pat of this expession exp( iu) υ() s = Re d π. (3.3) ( ) H ( + ) H ( ) The sign convention of the Dôme pape is that tailing paticles ae at u >. Leading paticles have u <, and, in this case the integation contou must be closed in the lowe half of the complex plane. Causality equies that the wakefield is eo fo leading paticles, and to satisfy this ) must be being teated as a complex quantity with a small, negative imaginay pat to avoid poles and cuts on the eal axis and ) thee be no poles in the lowe half plane. Figue A shows the complex plane fo ( ) B = ) ) ( ) + H H Thee is a cut along the negative eal axis and poles in the uppe half plane, but thee ae no poles in the lowe half plane, and causality is satisfied. The eal pat of eq. (3.4) is an even function of x just below the eal axis, and the imaginay pat is an odd function. This is shown in Figue A. These ae the same symmeties as exp( iu ), and the integal can be pefomed fom to exp( iu) υ() s = Re d π. (3.5) ( ) H ( + ) H ( ) (3.4)

6 ARDB-378 Bob Siemann & Alex Chao /4/5 Page 6 of 8 8 Integal Agument, =, = x + i*y y x Figue A: Contous of the magnitude of eq. (3.4) in the complex plane Integal Agument, = x.i Re Im Asymp Re Asymp Im x Figue A: The Re and Im pats of the agument of eq. (3.4) evaluated just below the eal axis. The appoximate expessions using eq. (3.6) ae also plotted.

7 ARDB-378 Bob Siemann & Alex Chao /4/5 Page 7 of 8 Expansions at Small u The asymptotic expession fo the atio of Hankel functions in the lowe half of the complex plane as is ) ) H 3 = i +. (3.6) H Using the lowest ode tem the denominato has eos at i ( + ) =, and =. (3.7) Eq. (3.3) is evaluated fo tailing paticles by closing the integal in the uppe half plane and using the Residue Theoem to give ( + ) u s υ() s = exp u =. (3.8) + R The tansvese wake is independent of. This could seem a poblem fo =, but in that case the limit leading to (3.6) is not satisfied. Keeping the next tem in (3.6) gives eos at ( ) + + +, = i ± = i ± E, (3.9) which ae both in the uppe half plane, and this equation defines E. Using these expessions and closing the contou integal in the uppe half plane and evaluating the esidues gives ( + ) u υ( u) = exp ( exp( ue) exp( ue )) E. (3.) + s s + u u = R R Exact Expession The exact expession, eq. (3.5), can be evaluated numeically by integating along the positive eal axis. The algoithm used is: ) Use the limiting fom of the Hankel functions at small to give 3 ) ) ( ) ( + ) H 4( + ) H ) Use the asymptotic expansion in eq. (3.6) fo > (3.) 3) Integate fom to the multiple of π neaest ( ) u π. The esults of the exact calculation and the appoximate expessions ae shown in Figue A3. The appoximate expessions ae seen to be good out to u =. (whee the calculation was stopped).

8 ARDB-378 Bob Siemann & Alex Chao /4/5 Page 8 of Tansvese Wakes fo Fo Feq. Indep. = exact lowest ode appox next ode appox. υ (u) u = s/r Figue A3: Tansvese wakefield fo e = independent of fequency. The exact expession, eq. (3.5), the lowest ode appoximation, eq. (3.8), and the next ode appoximation, eq. (3.), ae plotted. Bob Siemann & Alex Chao, "Wakefields in a Dielectic Tube with Fequency Dependent Dielectic Constant", ARDB-368 (4). G. Dôme, Poceedings of the Second Euopean Acceleato Confeence 68 (99). 3 M. Abamowit & I. A. Stegun, Handbook of Mathematical Functions, eq