The reciprocity theorem is used to prove the symmetry of the longitudinal impedance of an accelerating structure with respect to exchange of the coordinates of the leading and trailing particles. (2 refs)

Heifets, S

Zotter, Bruno W

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ON THE SYMMETRY OF THE IMPEDANCES. Heifets,y , B Zotter, CERNAbstractThe reciprocity theorem is used to prove the symmetryof the longitudinal impedance of an accelerating structurewith respect to exchange of the coordinates of the leadingand trailing particles.1 INTRODUCTIONThe longitudinal impedance Z(!; rt; rl) is a Fourier com-ponent of the longitudinal wake field generated by a lead-ing particle with transverse offset rland experienced by thetrailing particle with offset rtlagging behind the leadingparticle at the distance s.Current of a relativistic particle moving in z-directionwith transverse offset rl= (xl; yl) has only longitudinalcomponent j(r; rt; t) = e(r   rl)(s ct).We use Fourier harmonics defined as j(t) =Rj(!)e i!td!=2. Fourier harmonics of the current arej(r; rl) = (x  xl)(y   yl)ei!s=c: (1)The leading particle moving in an accelerator struc-ture and having charge e may excite in the structure thelongitudinal electric field E!(r; s; rl). The longitudinalimpedance is defined as integralZ(!; rt; rl) = 1eZdsE!(rt; s; rl)ei!s=c; (2)for the beam propagating inz directions, respectively.We want to proof that impedance is symmetric in respectwith exchange of the transverse coordinates of the leadingand trailing particles,Z+(!; rt; rl) = Z+(!; rl; rt): (3)For a particular case of the resistive wall impedance ofa straight pipe, the symmetry was noticed before[1] fromthe explicit form of the longitudinal impedance. We try toproof the theorem for an arbitrary structure but with mirrorsymmetry in respect with z !  z. The proof is based onthe reciprocity theorem and symmetry of EM fields gener-ated by particles in a symmetric structure.2 SYMMETRY OF THE LONGITUDINALIMPEDANCEThe reciprocity theorem relates EM fields (E(r; rt),H(r; rt)) and (E(r; rl), H(r; rl)) driven by two arbitrarycurrents j(r; rt) and j(r; rl) correspondingly.Work supported by Department of Energy contract DE–AC03–76SF00515.y Email: heifets@slac.stanfdord.eduThe reciprocity can be deduced directly from Maxwellequations[2] which give the following identity:divfE(r; rt)H(r; rl)  E(r; rl)H(r; rt)g = (4)Z0[j(r; rt)E(r; rl)  j(r; rl)E(r; rt)] (5)Cross here means vector product.Integral taken over the left-hand-side over the volume ofa beam pipe can be transformed in the surface integral. Onthe metallic walls of the beam pipe, the tangential compo-nents of electric fields are related to the tangential compo-nents of the magnetic fields E = H  n, where n is unitvector perpendicular to the wall,  = (1   i)(!=2c) issurface impedance, and ( !) =  i(!) is skin depth.The surface integral over beam pipe with ideal conduc-tivity ( ! 0) is zero.For a surface with finite conductivity, the surface integralZdV div(EtHl) =ZdSn:(EtHl) (6)= ZdS(Ht n)(Hl n) (7)is symmetric with respect to indexes t; l. Hence, the surfaceintegral over the left-hand-side of Eq. (4) is zero for thefinite conductivity as well.Consider now the surface integrals over the surfacesclosing the beam pipe volume at s ! 1 in the planeperpendicular to the z-axis. In the case of finite conduc-tivity of the walls, the integrals are zero because radiatedfields are absorbed in the walls. For ideal walls the integralZ1dSfE(r; rt)H(r; rl) E(r; rl)H(r; rt)g (8)is the difference of two integralsZ1dS[Ex(r; rl)Hy(r; rt) Ex(r; rt)Hy(r; rl)] (9) Z1dS[Ey(r; rl)Hx(r; rt)  Ey(r; rt)Hx(r; rl)]: (10)We can assume that the beam pipes at infinity are straightpipes and express transverse components of the fields in theTM waves in terms of the longitudinal components:Ex=iq2@Ez@x;Ey=iq2@Ez@y; (11)Hx=  ik2@Ez@y;Hy=ik2@Ez@x: (12)Here q is propagating constant, (!=c)2 = q2 + 2.0-7803-5573-3/99/$10.00@1999 IEEE. 1118Proceedings of the 1999 Particle Accelerator Conference, New York, 1999It is easy to see that both integrals are zero. Cancellationof the surface integrals for the TE waves can be seen in thesame way.Interference between TM and TE modes depends on po-sition of the flanges because, for a given frequency, thesemodes have different propagating constants. Therefore, theinterference term can be put to zero by proper choice of theflanges location.Hence, integration of Eq. (4) givesZdV [j(r; s; rt)E(r; s; rl)  j(r; s; rl)E(r; s; rt)] = 0:(13)This is correct for arbitrary currents. For the currentsj(r; rl) propagating in z-direction, and j(r; rt; s) in z di-rection j(r; s; rt) =  (r   rt)e i!s=c, Eq. (9) gives ZdsE+!(rt; s; rl)e i!s=c=ZdsE !(rl; s; rt)ei!s=c:(14)Expressions in this equation are the same as those in def-inition of impedance. Hence,Z+(rt; rl; !) = Z (rl; rt; !): (15)Now we have to relate impedances Z  and Z+.To do this, let us compare solutions E(r; s; r0) of thewave equation driven by currentsj(r; r0) = (r   r0)ei!s=c; (16)with the same transverse offsets but moving in oppositedirection along the z-axis. The boundary conditions onthe beam pipe are the same in both cases but two so-lutions E(r; s; r0) have different asymptotics. For z-components, for example,E+(s)! D(!)eiq!s; s!1; (17)E+(s)! A(!)eiq!s+B(!)e iq!s; s!  1 (18)andE (s)!  [A(!)e iq!s+B(!)eiq!s]; s!1 (19)E (s)!  D(!)e iq!s; s!  1: (20)Hence, wave equations, boundary conditions and asymp-totics are the same for the fields E+(s) and E ( s).Therefore, z-components are related:E (r; s; r0) =  E+(r; s; r0): (21)Using this result and the definition ofZ  Eq. (2), we obtainZ (rl; rt; !) =  ZdsE !(rl; s; rt)ei!s=c; (22)orZ (rl; rt; !) =ZdsE+!(rl; s; rt)ei!s=c: (23)Expression in the right-hand-side after change of the signof integrand s coincide with Z+, See Eq. (2). Hence, for amirror symmetric structure, we getZ (rl; rt; !) = Z+(rl; rt; !): (24)and Eq. (11) gives finallyZ+(rt; rl; !) = Z+(rl; rt; !): (25)The symmetry means that beam-pipe radii at1 are equal.The self-field of the beam has only transverse componentsand does not contribute to the right-hand side.This conclude the proof.2.1 References3 REFERENCES[1] A. Piwinsky, “Impedances in lossy elliptical vacuum cham-bers“, DESY 94-068, April 1994[2] L.D. Landau, E.M. Lifshitz, “Electrodynamics of ContinuousMedia,“, $69, Nauka, 19571119Proceedings of the 1999 Particle Accelerator Conference, New York, 1999

On the Symmetry of the Impedance

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