Bane, Karl Leopold Freitag

Yokoya, K

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THE LONGITUDINAL HIGH–FREQUENCY IMPEDANCEOF A PERIODIC ACCELERATING STRUCTUREK.L.F. Bane, SLAC, Stanford, CAK. Yokoya, KEK, Tsukuba, JapanAbstractIn many future collider and FEL designs intense, shortbunches are accelerated in a linear accelerator. For exam-ple, in parts of the Linac Coherent Light Source (LCLS)a bunch with a peak current of 3.4 kA and an rms lengthof 30 microns will be accelerated in the SLAC linac. Insuch machines, in order to predict the beam quality at theend of acceleration it is essential to know the short rangewakefields or, equivalently, the high frequency impedanceof the accelerating structure. R. Gluckstern[1] has derivedthe longitudinal, high-frequency impedance of a periodicstructure, a solution which is valid for a structure with asmall gap-to-period ratio. We use his approach to derive amore general result, one that is not limited to small gaps.In addition, we compare our results with numerical resultsobtained using a field matching computer program.1 INTRODUCTIONLet us consider the infinitely periodic, cylindrically sym-metric structure depicted in Fig. 1. R. Gluckstern has de-rived the high frequency behavior of the impedance of sucha structure, to order (kg)−1=2 relative to the leading term,with k the wave number and g the gap, as[1]ZL(k) =iZ0ka21 + (1 + i)Larkg−1; (1)with  = 1. Note that ZL is the average impedance per unitlength (averaged locally over frequency to give a smoothfunction, and averaged over a distance in the structure largecompared to the period L), Z0 = 120Ω, and a is theiris radius. Gluckstern’s result was meant to be valid forg=L  1. In this report, following Gluckstern’s method,we will show that Eq. 1 is still valid in the general case g=Lnot small, but with  a function of g=L.Other authors have investigated the high frequency be-havior of the same structure. Their results agree in theleading order term (iZ0=ka2), but not in the constant in the higher order term. E. Keil, describing the so-called Sessler–Vaynstein optical resonator model of highfrequency impedance, obtains a constant  = 0:67[2] andS. Heifets and S. Kheifets give  = 8=  2:55[3]. G. Stu-pakov, considering the limiting case of a structure with in-finitesimally thin irises (i.e. g=L = 1), finds that, in thiscase,   0:46[4].Work supported by Department of Energy contract DE–AC03–76SF00515.Figure 1: Two cells of the geometry under consideration.2 ANALYTICAL STUDYLet us begin by briefly summarizing Gluckstern’s method:He divides the geometry of Fig. 1 into two regions, the piperegion (r  a) and the cavity region r  a. The fieldsare expanded in terms of Bessel functions in the pipe re-gion and in term of the cavity eigenfunctions in the cavityregion with the (perfectly conducting) metallic boundarycondition on the iris surface r = a. He obtains a relation(Eq. 2.14 in the first of Ref. [1]) between the azimuthalmagnetic field and the axial electric field along r = a.Then, by matching the fields in the pipe and cavity regionsalong r = a, he obtains an integral equation for the axialelectric field along r = a (normalized by Z0I0=ka2e−ikz ,with I0 the arbitrary driving current), F (z) (we follow hisnotation):Z g0dz0" bKc(z; z0) + 1Xm=−1bKp(mL+z0−z)#F (z0) = −i:(2)The kernels in this equation are those of the cavity regionbKc and of the pipe region bKp, with the bKc term and them = 0 term involving bKp giving the contribution to theimpedance of the cell that includes the point z, and the m 6=0 terms giving the contribution of the other cells. Once weknow F (z) the impedance is simply given byZL(k) =Z0ka2LZ g0dzF (z) : (3)For our purposes we do not need to know the details ofthe kernels bKc and bKp, but only their high frequency be-havior. Gluckstern found that the high frequency behaviorof the cavity kernel bKc is independent of the details of thecavity shape, and is given bybKc(z; z0) = − (1 + i)papk(z − z0)(z − z0) ; (4)where (z) is the step function (0 for z < 0 and 1 forz  0). The kernel in the pipe region bKp is given by0-7803-5573-3/99/$10.00@1999 IEEE. 1725Proceedings of the 1999 Particle Accelerator Conference, New York, 1999bKp(z) = −2iaeikz1Xs=11bseibsjzj=a ;bs = (k2a2 − j2s )1=2 ; [=m(bs)  0] ;(5)with js the s-th zero of the Bessel function J0. Gluck-stern shows that, at high frequencies, the m = 0 pipe ker-nel, bKp(z0 − z), is equal to the cavity kernel bKc(z; z0),as given in Eq. 4. As for the m 6= 0 terms in bKp,Gluckstern points out, that those with m > 0 oscil-late rapidly at high k, and therefore do not contribute tothe average impedance. The sum over negative m givesKs(v) −1Xm=−1bKp(mK + z0 − z)=2ia1Xs=11bsei(−s)v1− ei(−s)v ;(6)with v = (z0 − z)=L,  = kL, s = bsL=a. Noting thatup to order k−1=2, ( − s) can be replaced by j2sL=2ka2,this becomesKs(v) = −4L1Xs=1"1j2s−iL2ka2 eij2s vL=2ka21− eij2s L=2ka2 +2ka2ij2sL!#: (7)The first term in square brackets can be summed:P1=j2s = 1=4; the second term can be approximated byan integral:P1s=1 f(js)  (1=)R10f(x)dx, for k !1.Thus, we obtainKs(v) = −L"1 +r2Lka2ei=4G0(v)#; (8)withG0(v) =2pZ 10dx"ex2vex2 − 1 −1x2#=1Xn=11pn− v −1pn+ (12) ;(9)where  is the Riemann zeta function ((1=2) = G0(0) =−1:460). Note that it is the second term in the brackets ofEq. 7 that Gluckstern has let go to zero, which will accountfor the difference in his final result and ours.The original integral equation, Eq. 2, thus becomes(1− i)pkLZ g=L0dv0 G(v0 − v)F (v0)− iaLZ g=L0dv0F (v0) =aL; (10)withG(v) =2(−v)p−v + G0(v) : (11)This equation can be rewritten in the formZ g=L0dv0 G(v0 − v)F0(v0) = 1 ; (12)withF0(v) = F (v)=A ; (13)A =pkL(1− i)"aL+iaLZ g=L0dv F (v)#: (14)Note that Eq. 12, with F0(v) the unknown, contains onlyone parameter γ  g=L. The equation for the impedance,Eq. 3, becomesZL(k) =Z0ka2AZ γ0dv F0(v) : (15)Finally, the solution to this equation is an impedance of theform Eq. 1, with(γ) =pγZ γ0dv F0(v)−1; (16)a result that can be verified by substitution.We find that for small γ,  = 1+G0(0)=pγ+O(γ3=2),and for γ = 1,   1 = −G0(0)=  0:4648. Thenumerical solution of Eq. 12 gives  in general (see Fig. 2).A polynomial fit in pγ(γ) = 1− 1pγ − (1− 21)γ ; (17)given by the dashes in Fig. 2, agrees to within 1.5% withthe numerical result. Note that (0) is in agreement withthe result of Gluckstern, and (1) with that of Stupakov.Figure 2: The coefficient (γ). The dashed curve gives theanalytical fit, Eq. 17. The plotting symbols are numericalresults discussed in the next section.1726Proceedings of the 1999 Particle Accelerator Conference, New York, 19993 NUMERICAL COMPARISONTo confirm these results numerically we have used a fieldmatching computer program working in the frequencydomain[5] (see also Ref. [6]). For the geometry of Fig. 1and for a given k, this program matches the tangentialfields at r = a, and then performs normalizing integrals.The result is an infinite dimensional matrix equation thatis truncated and inverted to obtain ZL(k). In order to bet-ter study the asymptotic behavior, the program calculatesthe impedance along a path slightly shifted off the real kaxis, which has the effect of averaging and smoothing outthe many narrow impedance spikes otherwise found at highfrequencies. (Note that to obtain the short–range wakefieldfrom this impedance, after performing the inverse Fouriertransform, the result must also be multiplied by the fac-tor exp[Im(k)s], with s the distance between driving andtest particles.) As example geometry we consider that ofa typical cell of the NLC accelerating structure known asthe damped, detuned structure (DDS)[7]. One simplifi-cation in our model, however, is that the irises are notrounded, unlike those in the real structure. The dimensionsare a = 4:924 mm, g = 6:89 mm, and L = 8:75 mm(note that for the average, high–frequency impedance nei-ther the cavity radius b, nor the coupling manifolds thatcouple through slots at r = b in the DDS, play a role).The numerical results, giving the real (RL) and imag-inary (XL) parts of the impedance ZL, when Im(k) =0:5 mm−1, are given in Fig. 1 (the solid curves). We notethat this impedance is indeed a relatively smooth functionof Re(k) (on the real axis, RL would be a collection ofmany infinitesimally–narrow spikes). We should point out,however, that with this method, to get good convergence inthe solution at high frequencies, the size of the matrix thatneeds to be solved becomes very large: at kR = 200 mm−1its size is  600  600. In Fig. 3 the dashed curves givethe analytical result, Eq. 1 with  = 0:52. We see thatagreement is good for frequencies ka & 1:5.The impedance was calculated for several values of g=Lwith the field matching program, keeping the other dimen-sions fixed. The results, for fixed Im(k), become bumpieras g=L decreases, due to the shorter reflection time 2g=cof a wave between irises; and for the calculations we letIm(k)  2=g. Then to obtain , the numerically obtainedRe(1=ZL) was fit to the form f(k) + g(k) (with f and gknown functions taken from Eq. 1). The results of the fitare given in Fig. 2 (the plotting symbols). The agreementis relatively good, though, due to residual bumpiness in theimpedance curve, the accuracy of the fit to the numericalresult is limited to  10%.Figure 3: The real (RL) and imaginary (XL) parts of theimpedance for the dimensions of the average NLC cell, asobtained by field matching (solid lines). The dashes repre-sent Eq. 1 with  = 0:52.4 CONCLUSIONSWe have extended R. Gluckstern’s analytical result for thehigh–frequency, longitudinal impedance of a periodic ac-celerating structure, a result valid for small gaps, to onevalid for all gap to period ratios. We have, in addition,performed numerical calculations and obtained results thatconfirm the analytical result to an accuracy of 10%.5 REFERENCES[1] R. Gluckstern, Phys. Rev. D 39, 2773 (1989); and R. Gluck-stern, Phys. Rev. D 39, 2780 (1989).[2] E. Keil, Nucl. Instr. Meth. 100, 419 (1972).[3] S. Heifets and S. Kheifets, Phys. Rev. D 39, 960 (1989).[4] G. Stupakov, Proc. of IEEE Part. Acc. Conf., Dallas, 1995,p. 3303.[5] K. Yokoya, KEK Report 90-21, September 1990, p. 142-150.[6] K. Bane, et al, “Calculations of the Short Range Lon-gitudinal Wakefields in the NLC Linac,” Presented atICAP’98, Monterrey, CA, Sept. 1998, and SLAC-PUB-7862(Revised), Nov. 1998.[7] “Zeroth-Order Design Report for the Next Linear Collider,”SLAC Report 474 (1996).1727Proceedings of the 1999 Particle Accelerator Conference, New York, 1999

The Longitudinal high frequency impedance of a periodic accelerating structure

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