PHYSICAL REVIEW SPECIAL TOPICS - ACCELERATORS AND BEAMS 13, 104401 (2010)
High-frequency impedance of small-angle tapers and collimators
G. Stupakov SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
B. Podobedov Brookhaven National Laboratory, Upton, New York 11973, USA (Received 13 July 2010; published 4 October 2010) Collimators and transitions in accelerator vacuum chambers often include small-angle tapering to lower the wakefields generated by the beam. While the low-frequency impedance is well described by Yokoya’s formula (for axisymmetric geometry), much less is known about the behavior of the impedance in the high-frequency limit. In this paper we develop an analytical approach to the high-frequency regime for round collimators and tapers. Our analytical results are compared with computer simulations using the code ECHO.
DOI: 10.1103/PhysRevSTAB.13.104401 PACS numbers: 41.60. m, 03.50.De, 41.20.Jb
I. INTRODUCTION b b b b ¼ arctan 1 2 1 2 1: The impedance of small-angle axisymmetric tapers with l l perfectly conducting walls was first computed analytically We assume that a beam propagates along the axis of the by Yokoya in the limit of low frequencies [1]. In this limit collimator at the speed of light. Our goal is to calculate the the longitudinal impedance is purely imaginary, which longitudinal impedance of the collimator. means that the beam does not lose energy to radiation. Later works [2,3] generalized Yokoya’s approach for rect- angular and elliptical cross sections of the transitions. In II. THE METHOD the opposite limit of very high frequencies, a so-called We will use a method of eigenmodes, in which the optical model has been developed [4,5] which predicts a electromagnetic radiation field of the beam is represented real longitudinal impedance. Simulations show, however, by a sum of modes of the empty waveguide. It is based on that there is a large range of frequencies between Yokoya’s calculation of the energy radiated by the image currents theory and the optical impedance where both theories fail induced by the beam in the walls of the waveguide. In the to provide an accurate result. In this paper we address this absence of other losses, the radiated energy is equal to the intermediate regime between the two limiting theories. energy loss of the beam and can be related to the real part This paper uses the method developed in an earlier paper of the impedance. The imaginary part of the impedance can by one of the authors [6], which attempted to solve this then be found using the Kramers-Kronig relations between problem, but failed to take into account the effect of mode the imaginary and real parts of the impedance. transformation in transition regions. The Fourier component of the beam current is (we In this paper we consider the geometry of an axisym- assume the e i!t time dependence in what follows) metric collimator shown in Fig. 1. It consists of two ¼ ikz identical conical tapers of length l connected by a section I! I0e ; (2) of a cylindrical waveguide of length g. The radius of the pipes outside of the collimator is b1, and the radius of the where ! stands for frequency, I0 is the amplitude of the ¼ pipe between the tapers is b2. We use cylindrical coordi- current harmonic, and k !=c. Let us denote the time- nate system r; z; with the origin of the coordinate z averaged intensity of radiation of this current from the situated in the middle of the collimator. The system is collimator region by P!. The real part of the impedance then symmetric with respect to reflection in the plane z ¼ is then given by the following relation (see, e.g., [8]): 0 [7]. The radius of the collimator bðzÞ as a function of z is defined by g 8 l l > j j g < b2; 0 < z < 2 ; bðzÞ¼ þð Þ jzj g=2 g j j þ g (1) > b2 b1 b2 l ; 2 < z
1098-4402=10=13(10)=104401(6) 104401-1 Ó 2010 The American Physical Society G. STUPAKOV AND B. PODOBEDOV Phys. Rev. ST Accel. Beams 13, 104401 (2010)
2P of position of the cross section. The right-hand side of Re Zð!Þ¼ ! : (3) 2 Eq. (8) can be considered as a scalar product of the two I0 þ þ fields En ; Hn and En ; Hn . One can show [9] that the The radiation is due to the image currents induced in the scalar product of two different modes is equal to zero, so perfectly conducting walls in the taper regions where the that a generalization of Eq. (8) for n j is walls are not parallel to the z axis. It is convenient to c Z m ðEþ H E HþÞ ¼ represent the total electric field of the beam current (2) n j j n dS Nn nj: (9) inside the taper as a sum of the vacuum field, Evac, and the 4 radiation field Erad, E ¼ Evac þ Erad, where for an on-axis We will use this equation in the next section. beam The energy radiated by the current P! can be written as a 2I sum over all possible modes, E vac ¼ r^ 0 eikz; (4) X 2 rc P! ¼ Pnjanj ; (10) with r^ being a unit vector in the radial direction of the n cylindrical coordinate system. where Pn is the energy flow in the mode of unit amplitude. The radiation field Erad satisfies Maxwell’s equation with the boundary condition that requires the tangential III. EIGENMODES IN THE COLLIMATOR component of the total electric field on the wall to vanish As described in the previous section, to calculate the E radj ¼ Evacj t wall t wall: (5) excitation of electromagnetic field by the beam, one needs to know the eigenmodes of the complete circuit. Analytical It is convenient to replace the boundary conditions (5) for a expressions for eigenmodes are available for cylindrical nonvanishing tangential component of the radiation elec- and conical waveguides; however, there is no a compact imag tric field on the wall by the surface magnetic current expression for eigenmodes of a collimator shown in Fig. 1. located inside the waveguide infinitesimally close to the More precisely, a single conical mode that propagates in wall [9,10]. The magnitude and direction of the magnetic the left taper of the collimator experiences transformation current is given by at the transition to the straight central section, generating c c several modes in the central part. Each of these modes, in i mag ¼ n Eradj ¼ n Evacj ; (6) 4 t wall 4 t wall turn, experiences a transformation at the second transition from the cylindrical waveguide to the right taper, resulting where n is the unit vector normal to the surface and in multiple conical modes in the right taper. We will have to directed towards the metal. Note that the magnetic current take these transformation processes into account in our exists only inside the tapers and vanishes in the region analysis. where the wall is parallel to the z axis. For calculation of the longitudinal impedance, one only Inside the waveguide, the radiation field excited by the needs axisymmetric TM modes. In the cylindrical central magnetic currents can be represented as a sum of eigen- part of the collimator, the modes propagating in the posi- modes, Eþ Hþ X tive direction, n ; n , are given by the following equa- rad þ E ¼ anEn ; (7) tions: n 2 þ jn r ð Þ þ ¼ i n z where a is the amplitude and E is the electric field of the Ez;n 2 J0 jn e n n b b2 nth eigenmode, propagating in the positive direction of the 2 þ ijnkn r ð Þ ¼ i n z (11) z axis. A similar expansion in terms of the amplitudes an is Er;n J1 jn e also valid for the magnetic field. Note that, in general, the b2 b2 sum in (7) also includes modes E propagating in the þ i!jn r ð Þ n ¼ i n z H ;n J1 jn e ; backward direction [9]. However, in the limit of high b2c b2 frequency, the modes that make a dominant contribution ¼ to the impedance propagate in the forward direction, and where n is the mode index, n 1; 2; ..., J0, and J1 are the ¼ the backward propagating modes can be neglected. Bessel functions, jn is the nth root of J0, and kn ð 2 2 2 2Þ1=2 The norm Nn of the mode n is defined as ! =c jn=b2 . The phase of the mode is equal to Z nðzÞ¼knz. The modes propagating in the negative di- c þ þ E H N ¼ m ðE H E H ÞdS; (8) rection, n ; n , are obtained from the forward modes by n 4 n n n n þ changing the signs of n and Er;n in (11). where the integral is taken over the cross section of the A simple calculation gives the norm (8) of the mode n: m waveguide and the unit vector is perpendicular to the N ¼ 1!k j2J2ðj Þ; (12) integration surface and points in the direction of propaga- n 2 n n 1 n tion. It can be shown that Nn does not depend on the choice with the energy flow in the mode equal to Pn ¼ Nn=4.
104401-2 HIGH-FREQUENCY IMPEDANCE OF SMALL-ANGLE ... Phys. Rev. ST Accel. Beams 13, 104401 (2010)
In the limit of high frequency, it turns out that only the IV. AMPLITUDES OF THE MODES AND MODE modes that propagate at small angles to the axis of the TRANSFORMATION system make the dominant contribution to the impedance Calculation of the mode amplitudes in the collimator is (so-called paraxial approximation, see [11]). In this ap- performed in several steps. At z ¼ ðl þ g=2Þ where the proximation one can neglect the transverse component beam enters the collimator, there is no radiation field j =b of the wave vector and replace k by k everywhere n 2 n present, hence a ¼ 0 in Eq. (7). in Eqs. (11) and (12), except for the phase . n n In the left taper, where there are magnetic currents (6), Analytical expressions for eigenmodes of the electro- the amplitudes a depend on z. The values of the ampli- magnetic field are also available for conical geometry (see, n tudes at the exit from the left taper, z ¼ g=2 (an e.g., [9]). In the general case of arbitrary cone angle and infinitesimally small here indicates a location right be- arbitrary frequency !, they involve the Legendre and fore the exit from the taper), are given by the following Bessel functions. We will use here a simplified version of integrals [9]: these functions valid in the limit of a small angle and Z high frequency !, replacing kn by k where possible [see ð Þ 1 1 mag ~ explanation after Eq. (12)]. The derivation of these modes an ¼ i H n dS; (15) N is given in the Appendix. In this limit, the conical eigen- n left modes (which we mark by the tilde below) are similar to where Hn is the magnetic field of the nth eigenmode the cylindrical ones, and, in our cylindrical coordinate propagating in the negative direction, dS is an infinitesimal system, they can be written as follows: element of the surface area, and the integration covers the wall area in the left taper where the magnetic current 2 þ jn r ð Þþ 2 ð Þ resides. E~ ¼ J j ei n z ikr =2R z z;n 2 0 n b At the transition point z ¼ g=2, the modes will be b transformed from conical to cylindrical ones, and the þ ijnk r ð Þþ 2 ð Þ E~ ¼ J j ei n z ikr =2R z (13) r;n b 1 n b amplitudes of the modes will be linearly transformed ð1Þ ð2Þ from an at z ¼ g=2 to an at z ¼ g=2 þ , þ i!jn r ð Þþ 2 ð Þ ~ ¼ i n z ikr =2R z H ;n J1 jn e ; X bc b ð2Þ ¼ ð1Þ an Snjaj : (16) j where the phase n is now determined from the differential 2 2 2 2 1=2 equation d n=dz ¼½! =c jn=bðzÞ . The factor We will discuss below how the matrix elements Snj are RðzÞ in the above equations is the curvature radius of the computed. spherical wave fronts of the modes in the conical regions; it The next mode transformation occurs at the transition ð Þ¼½ 0ð Þ ð Þ 1 ð Þ 0ð Þ ð3Þ is equal R z arctanb z =b z b z =b z . Note z ¼ g=2 with the new amplitudes an , ð Þ 0 ¼ that, due to the linear dependence of b z in the tapers, b X const. The sign of R is important: it is negative in the left ð3Þ ¼ ð2Þ an Rnjaj : (17) taper, corresponding to converging wave fronts of the j modes, and is positive in the right taper, where the wave fronts are diverging from the center of the collimator. Finally, radiation of the magnetic currents in the right taper Because Eqs. (13) differ from Eqs. (11) only by a phase ð3Þ will add to an : factor, the norm for the conical modes is the same as for the ¼ Z cylindrical ones, given by (12). The relation Pn Nn=4 ð Þ ð Þ 1 4 ¼ 3 imag H~ holds as well. an an n dS: (18) Nn right For the phase nðzÞ in (13) we have ð4Þ Z The amplitudes an are the final values that should be used z !2 j2 1=2 ðzÞ¼ n dz0: (14) in Eq. (10) to calculate the radiated power. n 2 ð 0Þ2 ð1Þ 0 c b z We now use Eqs. (4), (6), (12), and (13), to compute an in (15) The conical modes propagating in the negative direction Z ð1Þ 2iI0 g=2 dz ð Þþ ð Þ are obtained from (13) by changing the signs of n and ¼ ikz i n z ikb z =2 þ an ð Þ ð Þ e ; Er;n. ckjnJ1 jn l g=2 b z The small angle of the collimator, as was pointed out in (19) [6], allows one to neglect the reflection of eigenmodes at the transitions between the cylindrical and conical regions. where we have used the small-angle approximation, However, it does not preclude mode transformation at 1, and b0 in the left taper. Taking into account the these transitions, and we will account for this below. symmetry of the collimator and performing a similar cal-
104401-3 G. STUPAKOV AND B. PODOBEDOV Phys. Rev. ST Accel. Beams 13, 104401 (2010) culation for the second term on the right-hand side of 20 Eq. (18), one finds that it is equal to the complex conjugate 10 of (19). 0 To find the matrix elements Snj in Eq. (16), we note that 10 ¼ due to the continuity of the field at z g=2 it can be 20 ECHO expanded into the conical eigenmodes as well as the cy- WVpC 30 Present theory lindrical modes: 40 X X ð2ÞE~þ ¼ ð1ÞEþ an n aj j ; (20) 0.0 0.2 0.4 0.6 0.8 1.0 n j s cm with a similar continuity equation holding for the magnetic FIG. 3. Wake potential of a ¼ 100 m bunch. fields. Using the orthogonality property of the modes with z respect to the scalar product (9), we find While our algorithm directly finds only the real part of X Z impedance, we can find the imaginary part by making use ð Þ c ð Þ a 2 ¼ a 1 m ðEþ H~ E~ HþÞdS; of causality, which relates imaginary and real parts of the n 4 N j j n n j n j impedance via the Hilbert transforms (Kramers-Kronig (21) relations) [8]. To proceed, we need to define Re Z for all frequencies, so we set it to zero below fc, and set it equal to which defines the matrix elements in Eq. (16). The matrix the optical model value above fmax. Im Z calculated by the elements Rnj in Eq. (17) could be found in a similar Hilbert transform of this Re Z is shown in Fig. 2 in solid fashion. red. Below the cutoff frequency it ends up very close to the Yokoya value. For comparison, we plot impedances calcu- ¼ V. NUMERICAL RESULTS lated from a Fourier-transformed wake potential of a z 20 m Gaussian bunch computed by the EM code ECHO The impedance calculation algorithm described above based on a finite integration technique [13]. Running ECHO was implemented in MATHEMATICA [12]. In numerical we usually used the grid size from 5 to 200 times smaller calculations, the infinite sums involved in Eqs. (10), (16), than z and made sure that the numerical result converged and (17) are truncated, and only first Nm 10 lowest and did not change with further refining of the mesh. One modes are used to calculate the impedance. By varying can see a very good agreement between our approach and Nm we verified that the result does not depend on the exact the ECHO results. value of Nm. Since our algorithm allows one to accurately find the To illustrate our method, we have chosen the following impedance over a very broad frequency range, we can use ¼ ¼ ¼ ¼ collimator geometry: l g 3cm, b1 2b2 0:5cm, inverse Fourier transform to reconstruct the wake potential so that the collimator angle is ¼ 4:7 degrees. The real of a short bunch. For instance, for a Gaussian bunch with part of the impedance computed from the beam pipe cutoff, rms length ¼ 100 m, we obtain the wake potential ¼ ¼ ¼ z fc j1c=2 b2 46 GHz, up to the frequency fmax shown in Fig. 3, plotted with the ECHO result for compari- 3:9 THz is shown in Fig. 2 in solid blue. At higher fre- son. Again, we observe a perfect agreement between the quencies this impedance approaches the optical model two. ¼ð Þ ð Þ¼ value, Zopt Z0= log b1=b2 83 . Finally, in Fig. 4 we present the loss factor and the maximum absolute value of the wake potential as a func-
104 Present theory 100 ECHO pC V x VpC 1 ma loss
k Present theory
0.01 ECHO Ws
10 4 10 4 0.001 0.01 0.1 1 10
FIG. 2. Collimator impedance: present theory is shown with z cm solid line, the result of simulations with ECHO are shown with dots. FIG. 4. Loss factor and maximum of the wake potential.
104401-4 HIGH-FREQUENCY IMPEDANCE OF SMALL-ANGLE ... Phys. Rev. ST Accel. Beams 13, 104401 (2010) tion of bunch length. As we expect from the optical model, The transformation from the spherical coordinate system to 1 for short bunches, both quantities scale as z , while in the the cylindrical one r; ; z employed in the main body of j ð Þj / 2 ¼ ¼ opposite, Yokoya regime, W s max z and the loss our paper is given by z cos and r sin , where z becomes exponentially small. In the intermediate region is now measured from the vertex of the cone. [roughly 2 magnitude orders in z with corresponding We are interested in axisymmetric (no dependence) j ð Þj changes of 3 or more orders in magnitude in W s max TM eigenmodes that can propagate in the waveguide. and kloss] the scaling is more complex, and, to our knowl- These modes have three components of the field: E , E , edge, it is not described by any existing analytical treat- and H . The modes propagating away from the vertex in ments. Our new approach comfortably fills this gap. the positive direction of (denoted by the superscript ‘‘þ’’) are given by the following equations [9]: VI. CONCLUSIONS ð þ 1Þ 1 @2Uþ Eþ ¼ Uþð ; Þ;Eþ ¼ ; In conclusion, we developed a novel analytical approach ; 2 ; @ @ to find the impedance of (small-angle) tapered collimators þ (A1) þ ¼ ik @U in axially symmetric geometry. Impedance can be found H ; ; over a very broad frequency range, from DC to high- @ frequency optical model limit, thus allowing one to recon- where struct the wake potential of short bunches. We note that this sffiffiffiffiffiffiffiffiffiffi algorithm is also applicable to convex (cavitylike) struc- k ð Þ Uþð ; Þ¼ H 1 ðk ÞP ½cosð Þ ; (A2) tures. Our general method can also be used for tapers with 2 þð1=2Þ large angle ; however, this would require working with exact eigenfunctions in the conical region, given by ð1Þ with H þð1=2Þ the Hankel function of the first kind of order Eq. (A1), instead of simpler expressions, Eq. (13). þ 1 2 . The eigenvalues are found from the requirement We plan to apply our method to 3D geometries in the of the zero tangential electric field at the metal surface, future. P ½cosð Þ ¼ 0. The eigenmodes propagating in the negative direction ACKNOWLEDGMENTS þ are given by (A1) with U replaced by U . The function ð1Þ This work was supported by the U.S. Department of U is given by (A2) in which H þð1=2Þ is replaced by the ð Þ Energy under Contracts No. DE-AC02-76SF00515 and Hankel function of the second kind H 2 . No. DE-AC02-98CH10886. þð1=2Þ In the limit of small angles the eigenvalues are large, ½ ð Þ ½ð þ and we can use the approximation P cos J0 APPENDIX: DERIVATION OF APPROXIMATE 1Þ 2 , where J0 is the Bessel function of zero order, and EIGENMODES FOR A CONICAL WAVEGUIDE ð þ Þ ð þ 1Þ2 approximate 1 2 . We then can write rffiffiffiffiffiffiffi Consider a conical waveguide with its associated spheri- 2 þ ð1Þ cal coordinate system , , and , such that the origin of E ¼ J ð Þ xH ðxÞ ; 2 0 2 the system is located at the vertex of the cone, see Fig. 5. rffiffiffiffiffiffiffi 0 The metal surface of the waveguide is located at ¼ , k ð Þ Eþ ¼ J ð Þ xH 1 ðxÞ (A3) and the electromagnetic fields occupy the region < . ; 1 2 rffiffiffiffiffiffiffi þ i k ð1Þ H ¼ J ð Þ xH ðxÞ; ; 1 2 ¼ ¼ þ 1 where x k and 2 and the prime denotes the derivative with respect to x. The boundary condition now ð Þ¼ takes the form J0 0, from which it follows that j ¼ n ; (A4)
where jn is the nth root of J0. From (A4) we see that indeed small means 1. For large values of x, one can use the asymptotic ex- pression [14] sffiffiffiffiffiffiffi
ð1Þ 2 ic ðxÞ FIG. 5. Spherical and cylindrical coordinate systems associ- H ðxÞ e ; (A5) ated with a conical waveguide. x
104401-5 G. STUPAKOV AND B. PODOBEDOV Phys. Rev. ST Accel. Beams 13, 104401 (2010) with The second term in these equations is associated with the quantity kr2=2R in (A8) because the coordinate z measured 4 2 1 c ðxÞ x þ : (A6) from the vertex is equal to the curvature of the wave fronts 2 4 8x defined in (13)asR. The sum of the first and the third term indeed satisfies Eq. (A9) if one takes into account Substituting (A4) and (A5) into (A3) we obtain dbðzÞ=dz ¼ . j2 r Eþ ¼ n J j eic ðxÞ ; 2 0 n b b ij k r Eþ ¼ n J j eic ðxÞ (A7) [1] K. Yokoya, CERN Technical Report No. SL/90-88, 1990. ; b 1 n b [2] B. Podobedov and S. Krinsky, Phys. Rev. ST Accel. Beams ij k r 10, 074402 (2007). þ ¼ n ic ðxÞ H ; J1 jn e ; [3] G. Stupakov, Phys. Rev. ST Accel. Beams 10, 094401 b b (2007). where in the derivative with respect to x we used an [4] G. Stupakov, K. Bane, and I. Zagorodnov, Phys. Rev. ST approximation c 0 1 and replaced b ¼ and r=b ¼ Accel. Beams 10, 054401 (2007). = (see Fig. 5), where b is the radius of the metal surface [5] K. Bane, G. Stupakov, and I. Zagorodnov, Phys. Rev. ST Accel. Beams 10, 074401 (2007). in the cylindrical coordinate system. The last two equations ¼ [6] G. Stupakov, SLAC Report No. SLAC-PUB-7039, 1995. are approximations to the exact equalities b sin and [7] Reflection symmetry is not required for our method to ¼ r=b sin = sin in the limit ; 1. work, but it is used in this paper for simplicity. To prove that Eqs. (A7) are identical to (13), we need to [8] A. W. Chao, Physics of Collective Beam Instabilities in show that the phase c given by (A6) can be also repre- High Energy Accelerators (Wiley, New York, 1993). sented as [9] L. A. Vainshtein, Electromagnetic Waves (Radio i svyaz’, Moscow, 1988), in Russian. kr2 c ¼ ðzÞþ ; (A8) [10] R. E. Collin, Field Theory of Guided Waves (IEEE, New n 2R York, 1991), 2nd ed. with [11] G. Stupakov, New J. Phys. 8, 280 (2006). [12] MATHEMATICA version 7.0, Wolfram Research, Inc., 2008. d j2 1=2 j2 [13] I. Zagorodnov and T. Weiland, Phys. Rev. ST Accel. n ¼ k2 n k n : (A9) dz bðzÞ2 2kbðzÞ2 Beams 8, 042001 (2005). [14] I. Gradshteyn and I. Ryzhik, Table of integrals, series, and Observing that x ¼ k ¼ kz= cos kzð1 þ 2=2Þ products (Academic Press, New York, 2000), 6th ed. kz þ kr2=2z we substitute it to (A6) and neglect 1 in comparison with 4 2 in the last term kr2 2 c ðxÞ kz þ þ þ const 2z 2x kr2 j2 kz þ þ n þ const 2z 2k 2 kr2 j2 kz þ þ n þ const: (A10) 2z 2k b
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