Geometrical Wake of a Smooth Flat Collimator G

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SLAC-PUB-14583 GEOMETRICAL WAKE OF A SMOOTH FLAT COLLIMATOR G. V. Stupakov Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309

Abstract x A transverse geometrical wake generated by a beam pass- ing through a smooth ﬂat collimator with a gradually vary- z ing gap between the upper and lower walls is considered. Based on generalization of the approach recently developed for a smooth circular taper [1] we reduce the electromag- Figure 1: Sketch of a smooth ﬂat collimator. The collimator netic problem of the impedance calculation to the solution extends from y = −h to y = h in the y direction. A heavy of two much simpler static problems – a magnetostatic and line shows the trajectory of a driving particle. an electrostatic ones. The solution shows that in the limit of not very large frequencies, the impedance increases with the ratio h/d where h is the width and d is the distance between the collimating jaws. Numerical results are pre- sented for the NLC Post Linac collimator.

1 INTRODUCTION In this paper we calculate the impedance of a ﬂat smooth Figure 2: Cross section of the collimator by a plane z = collimator schematically shown in Fig. 1. The collimator const and transverse currents ﬂowing in the collimator extends in the y direction from y = −h to y = h and is walls. bounded by perfectly conducting walls. The beam propa- gates in the z direction. The collimator upper and lower walls are given by the equation x = b (z),whereb (z) is of the test particle, x, y. 0 ( ) 1 a smooth function such that b z . Throughout this The wake w corresponding to a purely imagi- ( ) paper we assume that h b z . nary transverse impedance (1) is w (x, y, x0,z)= For a smoothly varying wall and not very high frequency icZx (x, y, x0) δ (z) . After the passage of the collima- 2 = / ≈ −1 such that kb

Work supported in part by US Department of Energy contract DE-AC02-76SF00515. Eq. (2) to analogous to the electric one by introducing the magnetic “potential” ϕm such that Hy = −∂ϕm/∂x0. Note that the ∆ (0) = −4 ( − ) ( ) x,yϕe πδ x x0 δ y , (3) derivative in this equation is taken with respect to the loca- 2 2 2 2 tion of the driving particle x0, rather than x. The potential where ∆x,y = ∂ ∂x + ∂ ∂y , and the superscript 0 ϕm satisfies the Poisson equation indicates the zero approximation to the potential. The solution to Eq. (3) with the zero potential boundary condition ∆ϕm =4πδ (x − x0) δ (y) , (8) is sinh2 πy +cos2 π(x+x0) (0) =ln 4b(z) 4b(z) with the boundary conditions ϕe π(x−x ) . (4) sinh2 πy +sin2 0 4b(z) 4b(z) ∂ϕm =0 | =0 In the next approximation, the potential will be given by ,ϕm y=h . (9) (0) (1) (1) ∂n x=b(z) ϕe = ϕe + ϕe ,whereϕe is a first order correction that satisfies the following equation Again, invoking a perturbation theory, in the zero approx- 2 (0) imation which neglects the z-dependence in the potential (1) ∂ ϕe ∆x,yϕ = − , (5) ϕm,wehave e ∂2z2 (0) (1) ∆x,yϕm =4πδ (x − x0) δ (y) . (10) with the boundary condition ϕe =0.Theso- x=b(z) (0) lution to Eq. (5) can be found explicitly using ϕe as a The first of the boundary conditions (9) in this approxima- Green’s function, tion takes the form

(0) Zb(z) Z∞ ∂ϕm =0 (1) 1 0 0 . (11) ϕ (x, y, x0,z)=− dx dy ∂x e 4π x=b −b(z) −∞ 2 (0) The solution to this problem valid in the limit h b (z) is ×∂ ϕe ( 0 0 ; ( )) (0) ( − 0 0; ( )) 2 x ,y ,x0 b z ϕe x, y y ,x b z . (6) ∂z (0) 2 πy 2 π (x + x0) ϕm =ln 16 sinh +cos It is easy to show that the integral over y0 in Eq. (6) con- 4b (z) 4b (z) 0 verges on a scale of the order of |y |∼b which, by assump- 2 πy π (x − x0) πh × sinh +sin2 − . (12) tion, is much smaller than the half width of the collimator 4b (z) 4b (z) b (z) h. This observation justifies the limit h →∞assumed above. This function satisfies Eqs. (10) and (11), and is exponen- tially small at y = h, ϕm|y=h ∼ exp (−πh/4b).For 2.2 Magnetostatic Problem any practical purposes, the second of the boundary condi- 4 A specific feature of the magnetostatic problem is that even tions (9) can be considered as satisfied as soon as h> b. though h b (z), one cannot set h →∞and consider an Note that the presence of the second term on the right hand infinitely wide collimator. As we will see below, the con- side of Eq. (12) does not allow us to consider the limit h →∞in the magnetostatic problem. tribution of the magnetic field into impedance has a term (0) (1) that is directly proportional to the width of the collimator In the next approximation, ϕm = ϕm + ϕm ,where (1) h, and hence diverges in the limit h →∞. The physi- ϕm satisfies 2 (0) cal mechanism of this divergence is related to the currents (1) ∂ ϕm ∆x,yϕ = − . (13) generated in the walls of the collimator due to the variation m ∂2z2 of the image charges. This current flows around the colli- (1) mator cross section, as shown in Fig. 2, and generates the However, the boundary condition for the function ϕm dif- magnetic field Hy which turns out to be proportional to h. fers from the zero approximation (9), because the nor- It follows from the Maxwell equations that the y compo- mal to the wall n in the first approximation is equal to 0 nent of the magnetic field, Hy, of an infinitely thin current n =(1, 0, −b ), yielding the boundary condition wire between perfectly conducting walls satisfies the fol- (1) (0) lowing equation ∂ϕm ∂ϕm = b0 (z) . (14) 0 ∂x ∂z ∆Hy =4πδ (x − x0) δ (y) , (7) x=b x=b with the Neumann boundary condition at the upper and The solution of the inhomogeneous equation (13) with the lower walls, ∂Hy/∂n|x=b(z) =0, and the Dirichlet Neumann boundary condition (14) can be explicitly ex- (0) boundary condition Hy =0at the lateral walls y = h.It pressed in terms of the Green’s function ϕm [2] similar is convenient to formulate the magnetic problem in a way to Eq. (6). The result can be found in Ref. ([3]). 3 IMPEDANCE The impedance is given by Eq. (1),

∞ ! Z (0) (1) (0) (1) i ∂ϕe ∂ϕe ∂ϕm ∂ϕm Zx = − + − − dz. c ∂x ∂x ∂x0 ∂x0 −∞ (15) Note that from Eqs. (4) and (12) it follows that . . Figure 3: Schematic of the NLC collimator. (0) (0) ∂ϕe ∂x = ∂ϕm ∂x0, (16) which means that the zero order terms do not contribute to and the width of the collimator h =0.7 cm. The applica- the impedance. This fact can be easily explained by that bility condition of the theory developed in the previous sec- in the zero order theory the geometry of the collimator re- tions require the function b (z) to be smooth together with its first two derivatives. Strictly speaking, this requirement duces to the rectangular pipe of a constant cross section, in 0 which the wake of an ultrarelativistic beam is known to be does not hold for the profile shown in Fig. 3 where b (z) zero. In the first approximation we have, is not continuous at the entrance and the exit of the taper. In our calculations we assumed that in reality the angles ∞ ∞ ! Z Z (1) (1) of the collimator will be rounded in such a way that the ∂ϕe ∂ϕm (Ex − Hy) dz = − − dz. (17) smoothness condition is satisfied. ∂x ∂x0 −∞ −∞ The integrals (20) for this collimator are equal: I1 = −1 −2 0.08 cm and I2 =0.48 cm , which gives the trans- Near the axis when x, x0 b one can expand the gen- verse impedance near the axis, eral expressions for the impedance and carry out the in- 3 tegration over y. Omitting lengthy analytical calculations −ImZt =6.3 × 10 Ohm/m. (22) that were performed with the use of computer program Mathematica [4] we present here the result. We limit our 5 ACKNOWLEDGMENT consideration to the case y =0only, i.e. the case when both the driving and test particles are in the same vertical I would like to thank K. Bane, A. Chao, S. Heifets, J. Ir- plane. In this case, win, and R. Warnock for useful discussions. This work was supported by Department of Energy contract DE-AC03- −iZ0 76SF00515. Zx = (Ax + Bx0) , (18) 4π 6 REFERENCES where A =2I1,B= −2I1 +2πhI3, (19) [1] G. V. Stupakov. Geometrical Wake of a Smooth Taper. Par- ticle Accelerators, 56, 83 (1996). and Z 2 Z 2 [2] P. M. Morse and H. Feshbach. Methods of Theoretical (b0) (b0) I1 = dz ,I2 = dz . (20) Physics. McGraw-Hill, Inc., Chapter 7. b2 b3 [3] G. V. Stupakov. SLAC-PUB-7157, SLAC, 1996. For x = x0, for a conventional definition of the transverse [4] S. Wolfram. Mathematica: a System for Doing Mathematics impedance Zt,wefind by Computer. Addison- Wesley, Redwood City CA, 1991.

1 −iZ0 [5] K. Yokoya. Impedance of Slowly Tapered Structures. CERN Zt ≡ Zx|x=x = hI3. (21) x0 0 2 SL/90-88 (AP) (1990). [6] Zero Design Report for the Next Linear Collider, SLAC- Note that the integral I1 also appears in Yokoya’s theory of 474, SLAC, 1996. a smooth axisymmetric collimator [5], where b (z) plays a role of the pipe radius. Comparison of our result with [5] shows that the impedance of a ﬂat collimator is much larger (by factor of h/b ) than the impedance of a cylindrical collimator of a radius b (z).

4 NLC-TYPE COLLIMATOR We apply the results obtained above to collimators considered in the design of the Next Linear Collider [6]. The geometry of a typical collimator is shown in Fig. 3 with the following parameters: b =0.5 cm, g =0.1 cm, l =40cm,