Burton DA

Christie DC

Tucker RW

English

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IMPEDANCE OF ULTRARELATIVISTIC CHARGED DISTRIBUTIONS INTAPERING GEOMETRIESD.A. Burton, D.C. Christie, R.W. Tucker, Lancaster University and The Cockcroft Institute, UKAbstractWe develop a scheme for obtaining the impedance ofa gradually tapered, axisymmetric geometry containing abunch of arbitrary profile travelling at the speed of lightparallel to the axis of the taper. Coordinate-free expres-sions for Maxwell’s equations are 2+2-split in a coordinatesystem adapted to the particle beam and the taper and, us-ing an asymptotic expansion for a gradual taper, a coupledhierarchy of Poisson equations is obtained. Applications ofthe scheme are presented.INTRODUCTIONThe design of accelerator components such as collima-tors relies on understanding the consquences of passing anultrarelativistic charged beam through a waveguide with agradual taper. This is currently studied using a combinationof experiment and computer simulation. However, variousanalytical methods have also been developed to estimateimpedances (see, for example, [1]). Most recently, Stu-pakov [2] developed a process for evaluating the impedanceup to the second order of iteration for low frequency beamstravelling at v = c in perfectly conducting waveguides ofarbitrary cross-section. We shall use a similar method,restricted to axially symmetric confining geometries [3].However, our approach, using auxiliary potentials, enablesus to relax Stupakov’s low-frequency condition and pro-duce a hierarchy of equations that can be solved to arbitraryorder.MAXWELL EQUATIONS ANDBOUNDARY CONDITIONSThe spacetime metric g is given in cylindrical polar co-ordinates by1g = −dt⊗ dt + dz ⊗ dz + dr ⊗ dr + r2dθ ⊗ dθ (1)and the transverse, cross-sectional domainD at fixed t andz has the induced metricg⊥ = dr ⊗ dr + r2dθ ⊗ dθ (2)In spacetime, the Hodge map and exterior derivative aredenoted ? and d, and in the transverse domain, they aredenoted by #⊥ and d⊥. The transverse co-derivative δ⊥ isdefined asδ⊥ = #−1⊥ d⊥#⊥η (3)1We work in the MKS system, with units in which the speed of lightc = 1.where ηω = (−1)pω for any p-form ω and #⊥1 = rdr ∧dθ.The source, moving in the positive z-direction at the speedof light, has charge density ρ and 4-velocity fieldV = ∂t + ∂z (4)The vacuum Maxwell equations for the spacetime 2-formF are given bydF = 0, d ? F = − ρε0? V˜ (5)where V˜ = g(V,−) and ε0 is the permittivity of free space.In terms of new co-ordinatesu := z − t, ζ := z (6)the metric and volume 4-form ?1 = dt∧dz∧#⊥1 becomeg = dζ ⊗ du + du⊗ dζ − du⊗ du + g⊥ (7)?1 = dζ ∧ du ∧#⊥1 (8)and the velocity 1-form V˜ and its Hodge dual areV˜ = du, ?V˜ = du ∧#⊥1 (9)Taking the exterior derivative of the second Maxwell equa-tion implies that the charge density is independent of ζ andcan thus be written ρ(r, θ, u). One may uniquely expressF in terms of 0-forms Φ(r, θ, ζ, u) and Ψ(r, θ, ζ, u), and1-forms α⊥(r, θ, ζ, u) and β⊥(r, θ, ζ, u) which are inde-pendent of dζ and du:F = Φdζ ∧ du + du ∧ α⊥ + dζ ∧ β⊥ + Ψ#⊥1 (10)Furthermore, one may write [4]α⊥ = d⊥A + #⊥d⊥a, β⊥ = d⊥B + #⊥d⊥b (11)for 0-forms A(r, θ, ζ, u), a(r, θ, ζ, u), B(r, θ, ζ, u) andb(r, θ, ζ, u) provided A and B vanish on the boundary ∂D.This condition is compatible with the perfectly conductingboundary conditions that will be imposed on F below.Without loss of generality, it proves expedient to re-writethe form of F in terms of six new fields W , X , HB , Hb,HΦ andHϕ that will facilitate our subsequent analysis;A = ∂uW + ∂ζW −HB, B = HB − ∂ζWΦ = ∂uHΦ + ∂uHB + ∂ζHB − 2∂2uζW − ∂2ζζWa = ∂uX, b = ∂ζX −HϕΨ = ∂ζHϕ + ∂uHϕ +Hb − 2∂2uζX − ∂2ζζX (12)Proceedings of EPAC08, Genoa, Italy TUPP02605 Beam Dynamics and Electromagnetic Fields D04 Instabilities - Processes, Impedances, Countermeasures1589Thus, the Maxwell equations then reduce to the followingrelations:δ⊥d⊥HB = 0d⊥Hb = #⊥d⊥(∂uHB), d⊥Hϕ = #⊥d⊥HΦδ⊥d⊥W − 2∂2uζW − ∂2ζζW+ ∂uHΦ + ∂uHB + ∂ζHB = P (r, θ, u)δ⊥d⊥X − 2∂2uζX − ∂2ζζX + ∂ζHϕ + ∂uHϕ +Hb = 0(13)where ∂uP (r, θ, u) = ρ(r,θ,u)ε0 . The second equation in (13)implies the harmonic equationsδ⊥d⊥Hb = δ⊥d⊥Hϕ = δ⊥d⊥HΦ = 0 (14)The waveguide wall is the spacelike hypersurfacef := r −R(ζ) = 0 (15)for some smooth function R(ζ). We assume a perfectlyconducting boundary condition for F :df ∧ F = 0 at f = 0 (16)Equation (16) can be satisfied by settingW = ∂rX = 0, HB = ∂ζW, HΦ = −R′(ζ)1r∂θX(17)on the boundary f = 0.GRADUALLY TAPERING WAVEGUIDEConsider first a regular cylindrical waveguide with con-stant radius R(ζ) = R0. As ∂ζρ = 0, the source and con-fining geometry are both symmetric with respect to trans-lations in the ∂ζ direction. The simplest solution to theMaxwell system (13) with the boundary conditions (17) isthenX0 = Hb0 = HB0 = HΦ0 = Hϕ0 = ∂ζW0 = 0 (18)δ⊥d⊥W0 = P (r, θ, u) (19)with W0 = 0 on the boundary.A waveguide is defined to be gradually tapering iff := r − Rˇ(ζ) = 0 (20)where  is a small, dimensionless parameter. The fields willthen vary slowly with ζ. Introduce a ‘slow’ longitudinal co-ordinates = ζ (21)and rewrite all the potentials in terms of s, using the nota-tionχ(r, θ, ζ, u) = χˇ(r, θ, s, u) (22)where χˇ ∈ {Wˇ , Xˇ, HˇB, Hˇb, HˇΦ, Hˇϕ}. Express the po-tentials in the form of asymptotic series in :χˇ =∞∑n=0nχˇn (23)Note ∂ζχ = χˇ′ (where, from now on, a prime denotesdifferentiation with respect to s). The Maxwell equations(13) with boundary conditions (17) decouple to yield a hi-erarchical set of 2-dimensional Laplace and Poisson equa-tions for every order n, and the boundary conditions on HˇBnand HˇΦn depend on (n − 1)-order potentials. This leads toa straightforward procedure for calculating the potentialsorder-by-order. For n = 0, the only non-zero potential isWˆ0 which is a solution to δ⊥d⊥W0 = P (r, θ, u) and van-ishes at r = R(s).For every subsequent order of n:1. Calculate the harmonic potential HˇBn by solving the2-dimensional Laplace equationδ⊥d⊥HˇBn = 0 (24)subject to the boundary condition2HˇBn = Wˇ ′n−1 at r = Rˇ(s) (25)2. Calculate Hˇbn from 3d⊥Hˇbn = ∂u#⊥d⊥HˇBn (26)3. Calculate the harmonic potential HˇΦn by solving the2-dimensional Laplace equationδ⊥d⊥HˇΦn = 0 (27)subject to HˇΦn = −Rˇ′(s)1r∂θXˇn−1 at r = Rˇ(s)4. Calculate Hˇϕn fromd⊥Hˇϕn = #⊥d⊥HˇΦn (28)5. Calculate the potential Wn by solving the 2-dimensional Poisson equationδ⊥d⊥Wˇn = Wˇ ′′n−2 + 2∂uWˇ′n−1− ∂uHˇΦn − ∂uHˇBn − HˇB′n−1 (29)where Wˇn vanishes at r = Rˇ(s).6. Calculate the potential Xˇn by solving the 2-dimensional Poisson equationδ⊥d⊥Xˇn = Xˇ ′′n−2+2∂uXˇ′n−1−Hˇbn−∂uHˇϕn−Hˇϕ′n−1(30)with ∂rXˇn = 0 at r = Rˇ(s)2Throughout this section, we are dealing with the transverse Laplacianδ⊥d⊥. When considering the boundary conditions, s can thus be treatedas a parameter.3As HˇBn is harmonic, the converse of Poincare´’s Lemma guaranteesthat a solution exists to (26). Hˇbn is thus defined up to arbitrary functionsof s and u. These are subsequently constrained to zero by the boundarycondition on Xˆn. By an analogous argument, a unique value forHϕ canbe obtained fromHΦ using (28).TUPP026 Proceedings of EPAC08, Genoa, Italy05 Beam Dynamics and Electromagnetic Fields1590D04 Instabilities - Processes, Impedances, CountermeasuresEXAMPLEThe method can be used to replicate and extend thelongitudinal impedance4 calculation in [2] for a harmonicFourier component of a transverse delta-function beam off-set from the central axis. In our notation, the source termand impedance formula areρω(r, θ, u) = λωeiωu1rδ(r − r0)δ(θ) (31)Z‖(ω, r, θ, u) = −Z0 ε0λω∫ ∞−∞e−iωuΦdζ (32)= −Z0 ε0λω1∫ ∞−∞e−iωuΦˇds (33)where λω is the linear charge density, Z0 is the impedanceof free space and Φ is given by (12). First, Wˇ 0 is obtainedby solving δ⊥d⊥Wˇ 0 = − iωρω(r, θ, u) subject to Dirichletboundary condition at r = Rˇ(s). The solution isWˇ 0 = − iωp(u){ln(r2r20Rˇ(s)2 + Rˇ(s)2 − 2rr0 cos θ)− ln (r2 + r20 − 2rr0 cos θ)}(34)where p(u) := λωeiωu4πε0. Furthermore,Wˇ ′0 = −2iωp(u)Rˇ′(s)Rˇ(s){1 + 2∞∑m=1Υm(r, θ, s)}(35)where Υm(r, θ, s) :=(r0rRˇ(s)2)mcosmθ. Evaluating thepotentials according to the procedure in the previous sec-tion gives HˇB1 = W ′0, HˇΦ1 = Hˇϕ1 = 0 andHˇb1 = 4p(u)Rˇ′(s)Rˇ(s)∞∑m=1(r0rRˇ(s)2)msinmθ (36)Wˇ 1 =p(u)2Rˇ′(s)Rˇ(s)(Rˇ(s)2 − r2)[1 + ∞∑m=12Υm(r, θ, s)1 + m](37)Xˇ1 = p(u)Rˇ′(s)Rˇ(s)∞∑m=111 + m(r0rRˇ(s)2)m×(r2 − m + 2mRˇ(s)2)sinmθ (38)HˇB2 = p(u)Rˇ′(s)2[1 +∞∑m=12Υm(r, θ, s)1 + m](39)HˇΦ2 = 2p(u)Rˇ′(s)2∞∑m=1Υm(r, θ, s)1 + m(40)As can be seen from the equation for Φ in (12), Hˇb2,Hˇϕ2 , Wˇ 2 and Xˇ2 are not required in order to evaluate the4Transverse impedance can be obtained from the Panofsky-Wenzel re-lation [5].impedance to second order. The longitudinal electric fieldat this approximation isΦˇ = p(u){2Rˇ′(s)Rˇ(s)[1 + 2∞∑m=1Υm(r, θ, s)]+iω2[(Rˇ′(s)Rˇ(s)(r2 − Rˇ(s)2) [1+ ∞∑m=12Υm(r, θ, s)1 + m])′+ Rˇ′(s)2(1 + 4∞∑m=1Υm(r, θ, s)1 + m) ]}(41)The longitudinal impedance follows from (33). If thewaveguide aproaches constant radii R1 as s → −∞ andR2 as s → ∞, then R′(s) = 0 at s = ±∞ and the secondline of (41) will not contribute to the integral. Thus, to thisapproximation,Z‖ = Z0ε04πλω{2 lnR1R2− iω∫ ∞−∞Rˇ′(s)2ds+∞∑m=1(Υm∣∣∣R(s)=R2R(s)=R1− 4iω1 + m∫ ∞−∞ΥmRˇ′(s)2ds) }(42)After changing variable from s to ζ and truncating the se-ries at m = 1, the second-order impedance (42) is identicalto the tapered cylinder result in [2]. Evaluating Wˇ 2 andXˇ2 and repeating the procedure of the previous section forn = 3, 4, . . . yields higher order correction terms. Thethird order correction turns out to be zero for an asymptot-ically cylindrical pipe. The fourth order correction isZ‖4 =Z0ε04πλωiω3∫ ∞−∞(Λ1Rˇ′(s)4 + Λ2Rˇ′′(s)2Rˇ(s)2)ds(43)whereΛ1 =524+∞∑m=1Υmκ1m3(2m2 + 6m + 1− ω2(4m− 3)κ2mRˇ(s)2)Λ2 =324− ω212Rˇ(s)2 +∞∑m=1Υmκ1m(1− ω2κ2mRˇ(s)2)κ1m =2m(m + 1)(m + 2), κ2m =3m2 + 8m + 6m(m + 1)2(m + 3)while the fifth order contribution is zero.REFERENCES[1] Stupakov G V Part. Accel. 56, 83 (1996)[2] Stupakov G V, Phys. Rev. ST Accel. Beams 8 094401 (2007)[3] Tucker R W, Theoretical and Applied Mechanics, 34 (1):1-49(2007)[4] Abraham R, Marsden J E, Ratiu T Manifolds, Tensor Analysisand Applications (New York: Springer-Verlag, 1988)[5] Panofsky W K H, Wenzel W Rev. Sci. Instrum. 27 967 (1956)Proceedings of EPAC08, Genoa, Italy TUPP02605 Beam Dynamics and Electromagnetic Fields D04 Instabilities - Processes, Impedances, Countermeasures1591

Impedance of ultra-relativistic charged distributions in tapering geometries

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