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High-Frequency Impedance of Small-Angle Tapers and Collimators

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High-Frequency Impedance of Small-Angle Tapers and Collimators 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.Àm, 03.50.De, 41.20.Jb I. INTRODUCTION b À b b À 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 eÀi!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 À Þ jzjg=2 g j j þ g (1) > b2 b1 b2 l ; 2 < z <l 2 b1 b1 : g j j þ b2 b1; z >l 2 : 0 z Throughout this paper we assume that the angle of the collimator is small, FIG. 1. Geometry of an axisymmetric collimator. 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 À À 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 Þ j is walls are not parallel to the z axis. It is convenient to c Z m ðEþ  HÀ À EÀ  HþÞ ¼ represent the total electric field of the beam current (2) n j j n dS Nnnj: (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 ¼Evacj 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 in 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 in 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 À sum in (7) also includes modes E propagating in the þ i!jn r ð Þ n ¼ in 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 À 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 À À þ EÀ HÀ N ¼ m ðE  H À 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.
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