Electromagnetic field and short-range wake function in a beam pipe of elliptical cross section

(Received 18 April 2008; published 14 July 2008)Within the ultrarelativistic limit, analytical expressions are found for the high-frequency resistive-wallcoupling impedance of an elliptical cross-section vacuum chamber. Subsequently, the corresponding wakefunctions are derived by performing inverse Fourier transformations numerically. The electromagneticﬁelds have been developed working out two systems of solutions, namely for the vacuum and for theresistive wall. The constants involved in these systems have been determined by matching boundaryconditions at the interface vacuum wall. Several study cases have been considered concerning the aspectratio of the elliptical cross section and the transverse position of the leading charge in order to exemplifythe behavior of the longitudinal and transverse wake functions.


I. INTRODUCTION
Free electron laser (FEL) projects aim to achieve highbrightness photon beam pulses of minimum bandwidth. However, such pulses may be corrupted by possible large wakefields along the undulator small-gap vacuum chamber. Thus, knowledge of the short-range wakefields in the undulator vacuum chamber is needed to predict the beam quality in terms of the single bunch energy spread and emittance. A possible choice for a small-gap vacuum chamber is one with elliptical cross section, for which there are references to analytically derived expressions for the low-frequency resistive-wall coupling impedance (see for example [1][2][3]). Expressions for the highfrequency resistive-wall impedance are given, in the case of DC conductivity model, for a round pipe in [4], and for the conducting parallel plates in [5]. Adopting the AC conductivity model [6], the resistive-wall impedance is given for the round pipe in [7] and for the parallel plates in [8]. The problem of calculating the coupling impedance, including the high frequencies, of a resistive beam pipe with arbitrary cross section has been solved by Yokoya [9] with the boundary element method. He applied the method to numerically work out the solution in the case of an elliptical pipe; moreover, the method can be applied using both AC and DC conductivity models.
In this paper we analytically derive expressions for the longitudinal and transverse resistive-wall coupling impedance of a vacuum chamber with elliptical cross section using another method, precisely the field matching method, and make a comparison with the results obtained with the boundary element method on the same geometry. The paper is organized as follows. In Sec. II we describe the physical model used to obtain the expressions of the electromagnetic fields inside the vacuum and inside the resistive parts of the beam pipe, respectively. We use elliptical coordinates and write the field components in term of Fourier transformations. In Sec. III we work out the Maxwell equations to derive a series expression of the electromagnetic (e.m.) field in the vacuum part, while in Sec. IV we derive a series expansion of the e.m. field in the resistive part of the beam pipe. In Sec. V, the constants involved in the two series expansions are determined by imposing the boundary conditions at the interface vacuumresistive wall. This fully determines the field components. In Sec. VI we derive the expressions of the longitudinal and transverse resistive-wall impedances, and the short-range wake functions are then obtained by calculating numerically the inverse Fourier transformations of the impedances. In Sec. VII applications and examples are illustrated, involving different aspect ratios of the cross section, different materials using both AC and DC conductivity models, and different leading and trailing charges displacement from the beam pipe axis.

II. PHYSICAL MODEL DESCRIPTION
Let us consider an elliptical cross-section pipe with finite conductivity and infinite wall thickness, and denote with a the major half-axis of the ellipse and with b the minor one, respectively. Figure 1 shows cross section and longitudinal section of the beam pipe, and the traveling point charge with the elliptic cylindrical u; v; s coordinates system, where the s-axis coincides with the pipe axis. The relations between the Cartesian and the elliptical coordinates are given in Appendix A and are illustrated in Fig. 2. The equation u u 0 defines the surface separating the vacuum region from the resistive wall. The vacuum region is specified by u < u 0 while the metal region by u > u 0 . The leading point charge, traveling along the beam pipe, is assumed to be ultrarelativistic, and its longitudinal position is s ct, where c is the velocity of light in vacuum, while its transverse coordinates are u 1 ; v 1 .

III. FIELDS IN THE VACUUM
Using the elliptical cylindrical coordinates system Ou; v; s, the Maxwell's equations in the vacuum region can be written as follows, denoting the electric field components with E u , E v , and E s , and the magnetic field com-ponents B u , B v , and B s : where h is the metric while the charge and current density are In addition, we refer to the coordinate z s ÿ ct, which is the longitudinal displacement from the point charge. Thus it is z < 0 behind the leading charge and z > 0 ahead of it. Because of the causality principle, all fields must vanish for z > 0.
Using the same approach adopted in [4], we write the field vectors E E u ; E v ; E z and B B u ; B v ; B z in terms of the Fourier transformed vectorsẼ,B on the z-axis: (4c) c @hB u @u @hẼ u @v Starting from Eqs. (4a) and (4b), and following the approach described in Appendix B, the longitudinal components of the electric and magnetic fields can be written as follows: where A n and B n are constants to be determined. Starting from Eqs. (4c) and (4d) and using the solutions in (5), the approach described in Appendix C yields the following expressions for the transverse fieldsẼ u andB u : The field componentsẼ v andB v can be directly obtained from Eqs. (4e) and (4f) using the solutions (5) and (6), yielding

IV. FIELDS IN THE RESISTIVE WALL
The constants A n , B n , E n , and F n will be determined by imposing the boundary conditions at the surface separating the vacuum from the resistive wall. To this aim we need to calculate the expressions of the fields inside the wall (u > u 0 ). In the conductor we assume that 0; (10a) where is the conductivity of the metal. Furthermore, the Maxwell equations can be written as Using the variable z s ÿ ct and manipulating Eqs. (11) yields equations where 2 ikZ 0 and is chosen with positive imaginary part.

A. Longitudinal fields in the conductor
Each of Eqs. (12a) and (12b) can be solved by separating the variables u and v, that is assuming the unknown is proportional to the product between two functions Uu and Vv. Substituting the product UuVv where Q l 2 2 =4 and a is a separation constant. Equations (13a) and (13b) are called the Mathieu angular and radial equations, respectively [10]. On the other hand, the electric and magnetic fields, as well as the function Vv, must be 2-periodic in v. There is a countable infinity of values of the constant a that allow 2-periodic solutions in v. Such values are called the Mathieu characteristic numbers (MCNs), and can be calculated with the algorithms in [11]. In particular, for imaginary values of Q (i.e., in the case of DC conductivity), a useful algorithm can be found in [12].
Adopting the standard notation for the MCNs [13], the constants a 2n and a 2n1 (n 0) produce even -periodic and 2-periodic solutions, respectively, while the constants b 2n (n 1) and b 2n1 (n 0) produce odd -periodic and 2-periodic solutions, respectively.
Furthermore the possible solutions Vv can be written as where the Fourier coefficients can be calculated by recursion formulas [13].
It is worth noting that, for each of the solutions in (14), the other linearly independent solution of the Mathieu angular equation, for a fixed MCN, is not periodic in v [10], thus can be discarded.
The Mathieu radial equation has two linearly independent solutions for every MCN. They are called the first and the second kind radial Mathieu functions, respectively, and can be expressed as series of products of Bessel functions [11].
Since we are interested in damped solutions inside the conductor, a proper linear combination of the solutions of first and second kind has to be chosen. The damped solutions required for a complex Q are where J n is the Bessel function of the first kind of nth order, H 1 n is the Hankel function of first kind of nth order, w 1 Q p e ÿu and w 2 Q p e u , Q p having positive imaginary part. Being proportional to UuVv, the unknownsẼ z and cB z , can be finally expressed as where the constants C E a n , D E b n , C B a n , D B b n are to be determined.

B. Transverse fields in the conductor
The transverse fields in the conductor can be directly calculated by substituting Eqs. (16a) and (16b) into Eqs. (12c)-(12f). As in the following only tangential fields are involved, we give only the expressions for hẼ v and chB v : Phys. Rev. ST Accel. Beams 11, 074401 (2008) 074401-5 hẼ v X 1 n0 ÿ ik 2 C E a n U a n X 1 m1 mA a n m sinmv ÿ A a n m cosmv ÿ ik 2 C B a n U a n X 1 m1 mA a n m sinmv X

V. EVALUATION OF THE CONSTANTS
The constants involved in the field expressions are determined by satisfying the boundary conditions at u u 0 , separating the vacuum from the resistive wall.
Precisely , the continuity ofẼ z , hẼ v , cB z , and hcB v , the field components tangential to the surface u u 0 , is imposed. As a result, the constants A 2n , A 2n1 , B 2n , and B 2n1 can be calculated by solving through four independent tridiagonal linear systems. More details about the estimation of the linear system can be found in Appendix D. Truncating every tridiagonal linear system at the M 1th order we obtain Since the coefficients on the diagonals are not zero, the system can be easily determined by Gaussian elimination without pivoting. Thus, the following recursive formula can be used to obtain the coefficients: Introducing the quantities l a 2 ÿ b 2 p , W ikl 2 , 2 ikZ 0 , Ru 0 il coshu 0 , the coefficients s n , d n , z n , and t n on the diagonals are given by the following relations for each system:

VI. SHORT-RANGE WAKE FUNCTIONS
Let us consider an ultrarelativistic trailing charge q t traveling with a longitudinal displacement z from the leading charge q; its transverse coordinates are u; v. The Lorentz force experienced by the trailing charge due to the leading charge is The longitudinal and transverse components of F are where the fields E z , E u , E v , B u , and B v are evaluated in u; v; z and depend on the position u 1 ; v 1 ; 0 of the leading charge q. Using Eqs. (4e) and (4f); we can write Thus, the force depends only on the longitudinal components E z and B z . The wake function per unit of length is defined by W F=qq t , and is related to the impedance through the following Fourier transformation: Wz; u; ve ÿikz dz: Consequently, the longitudinal impedance per unit of length is given by 1=qc times the expression ofẼ z k in Eq. (5a): while the two components Z u and Z v of the transverse impedance Z T Z uû Z vv are calculated using Eqs. (27b) and (28): Thus, the wake functions are obtained by a numerical inverse Fourier transformation of the impedance. The fields vanish for z > 0 and since the real part of the impedance drops more quickly for large values of k than the imaginary part it seems to be more convenient to use the cosine inverse transformation:

VII. EXAMPLES AND RESULTS
In this section the impedances and the short wake functions are calculated for several cases, such as cross sections with different aspect ratio values a=b and two different kinds of conductors, aluminum, and copper with DC and AC conductivity values. The conductivity and the relaxation time used for aluminum and copper are listed in Table I. The relative displacement between the leading and the trailing charge is also considered. Basically, the tridiagonal linear system in Eq. (18) must be truncated and when we consider large values for the aspect ratio a=b, more sine and cosine components are needed to represent the fields accurately. A particular case is when both charges, leading and trailing, travel along on-axis in the vacuum chamber. In this case, only the A even subsystem is excited, and the longitudinal impedance is then simplified as In a small region around the axis, the longitudinal wake function remains approximately constant while the transverse wake function depends linearly on the displacements of the leading and the trailing charges. This effect is shown in Fig. 3. The longitudinal wake function has been calculated at z 0 with different transverse positions of both the leading and the trailing charges from the axis (offset). In the following examples we applied our method using vacuum chambers having the short half-axis length b 3 mm. The reason is that this value is equal to the half gap for the undulator vacuum chamber of the FERMI@Elettra FEL project [14]. Nevertheless to better compare our plots with the results obtained with the boundary element method by Yokoya, Fig. 6 has been obtained using b 1 cm, the DC conductivity model, and the same aspect ratio values presented in [9]. Figure 4 shows the longitudinal impedance with the AC conductivity model for copper and aluminum, as a function of the wave number k and for several values of the aspect ratio a=b.    Figure 5 shows the longitudinal wake functions for a vacuum chamber with semiminor axis b 3 mm and several aspect ratio values a=b, as a function of the longitudinal displacements behind the leading charge. It is worthwhile noting that the longitudinal wake functions reduce to those of the circular case when a=b 1 [7] and to those of the parallel plates when a=b 1 [8], respectively. Figure 6 shows the transverse wake functions obtained using DC conductivity model for copper and a semiaxis b 10 mm. It is worthy to mention that in the figure the conductivity used is 5:3 10 7 Sm ÿ1 and that the results obtained reduce correctly to the circular case [4] and to the parallel plates case [5]. Furthermore, the results are in good agreement with the solutions proposed in [9]. Figure 7 shows the transverse wake functions as functions of the longitudinal displacement behind the leading charge, for aluminum. Here we considered three different relative transverse positions between the leading and the trailing charges: both charges with a y offset, the leading charge off-axis and the trailing charge on axis and vice versa. The semiminor axis b is 3 mm and the more signifi-cant case of AC conductivity model is considered. Figure 7(f) shows that @ @x W x @ @x 1 W x vanishes for large values of a=b, when the ellipse approaches the parallel plates limit. When an ultrashort bunch is considered, then the transverse wake with the leading and the trailing charges off axis with the same offset should be considered. Figures 7(e) and 7(f) represent this case. As explained in [15], the transverse wake forces near the axis of a bisymmetric pipe (the elliptical pipe has mirror symmetry in both x and y) have the property @W y @y ÿ @W x @x . This explains why Fig. 7(d) is the mirror of Fig. 7(c).
Once the wake function is known, the energy change per unit length (in eV=m) induced within a particle bunch is obtained from the convolution: where e is the electron charge and Q is the total bunch charge with longitudinal charge distribution z expressed in m ÿ1 . Figure 8 shows the convolutions obtained with the wake functions and flattop bunches of charge equal to 1 nC and different lengths.
As for the longitudinal case, once the transverse wake function is known, the transverse kick for unit length and for transverse offset (in rad=m 2 ) received by the particles in   the vertical plane within the bunch is obtained from the following convolution: where e is the electron charge, Q is the total bunch charge with longitudinal bunch distribution z, and E 0 is the bunch mean energy. Here only the vertical plane is considered, because for aspect ratio values a=b > 3 the effect of the transverse wake for a horizontal offset can be neglected. Figure 9 shows the convolutions obtained with the wake functions and flattop bunches of charge equal to 1 nC and different lengths.

VIII. CONCLUSION
In this paper we have analytically derived expressions for the high-frequency longitudinal and transverse resistive-wall coupling impedance of an elliptical crosssection vacuum chamber. Then, the corresponding longitudinal and transversal wake functions have been obtained by calculating numerically the inverse Fourier transformations of the impedances. Once the longitudinal wake function was known, the energy changes per unit length induced within particle bunches was estimated by means of a convolution between the wake function and several flattop charge distributions. The results show that the energy variation induced within the bunch could assume an unacceptably large value when shorter electron bunches are used. Using the aluminum as material and an elliptical shape with aspect ratio a=b > 3 for the vacuum chamber, the amplitude and the number of oscillations can be reduced.
As for the longitudinal case, the transverse kick for unit length and for transverse offset received by the particles in the vertical plane within the bunch is obtained through the convolution between the transverse wake function and the flattop charge distributions. For the particle bunches considered, the results have shown that the kick angle received from the head and tail of the bunch itself maintains an acceptable value even for shorter lengths of the charge distributions. It is worthwhile mentioning that in a small region around the axis the longitudinal wake function remains approximately constant while the transverse wake function depends linearly on the displacements of the leading and trailing charges.

ACKNOWLEDGMENTS
We thank C. Bontoiu for discussions on the FERMI@Elettra undulator vacuum chamber and for carefully reading the manuscript.

APPENDIX A: ELLIPTICAL CYLINDRICAL COORDINATES
With reference to Fig. 2, the Cartesian coordinates x; y; s and the elliptical coordinates u; v; s are related as follows: x l coshu cosv; (A1) y l sinhu sinv; where l a 2 ÿ b 2 p is the half focal length. The metric coefficients for the elliptic cylindrical coordinates are h h u h v l cosh 2 u ÿ cos 2 v p : The inverse of (A1) and (A2) is The unit vectorsû andv are expressed as û x sinhu cosv ŷ coshu sinv The segment u 0 connects the foci, and the couples 0; v and 0; ÿv represent the same point on the cross section. Note that the unit vectorsû andv are discontinuous on the segment u 0. Precisely, where F 00 u denotes the second derivative on u. The latter equation is equivalent to the following system of infinitely many equations:  where gu; v is the generic forcing term. For each a particular solution is reported in Table II. Particular solutions related to the longitudinal fields are given by (7c)-(7f). Substituting the solutions KẼ u , KB u and (7c)-(7f) into the first order system of Eqs. (4c) and (4d), yields the conditions A 0u 0, B 0u 0, A n cD n , B n ÿcC n , E n ÿcH n ÿ ni k B n , and F n cG n ÿ ni k A n . Furthermore, the following conditions of continuity of the vectors hẼ uû and hcB uû must be imposed on the segment u 0: hB u 0; v ÿhB u 0; ÿv; (C6) hẼ u 0; v ÿhẼ u 0; ÿv: The latter conditions lead to impose: A 0 0, B 0 0, A n 0, and C n 0. In conclusion, the solutions KẼ u and KB u of the homogeneous Eqs. (C4a) and (C4b) are given by (7a) and (7b), respectively. The particular solutions of (C4a) and (C4b), related only to the charge q, are denoted with S q Eu and S q Bu , respectively, and are obtained by starting from the following Fourier series expansion of the u-components of the fields and of the source terms, as follows:

APPENDIX D: IMPOSING THE BOUNDARY CONDITIONS
To evaluate the constants in the fields expressions, the continuity of the tangential componentsẼ z ,Ẽ v ,B z ,B v on the surface u u 0 must be imposed.
Observing the Fourier expansion of the fields [Eqs. (7) and (9a)], it can be noticed that there is no relation between even and odd constants, because the nth component of the expansion depends on the n ÿ 2th and on the n 2th components in the vacuum, while Eq. (14) shows that angular Mathieu functions are sums of either -periodic