Impedances of Anisotropic Round and Rectangular Chambers

We consider the calculation of electromagnetic fields generated by an electron bunch passing through an anisotropic transversally non-homogeneous vacuum chamber of round or rectangular cross-section with translational symmetry in the beam direction. The described algorithms are implemented in a numerical code and cross-checked on several examples.

The anisotropic dielectrics may be incorporated either intentionally or unintentionally (processing-induced anisotropy) [4]. Dielectric permittivity and conductivity depend on the direction of wave propagation and polarization in many materials. The anisotropy can have a significant effect on modal coupling and must be accounted for in the design and analysis of such structures.
There are many papers which describe impedance calculations of steady-state impedance for isotropic round and flat layered chambers [5][6][7][8][9][10] with translational symmetry in the beam direction. The solutions for isotropic structures are obtained in analytical form or a field-matching approach can be used to reduce the problem to a simple matrix equation. In this paper we consider anisotropic transversally nonhomogeneous round and rectangular chambers where the field-matching technique does not work.
We start in Section II from formulation of the problem. Then we review the general form of the impedance for round and rectangular waveguides in non-relativistic case. For a special case of uniaxial anisotropy in the beam direction the same field matching approach as for the isotropic case [5] can be used. We consider shortly the required modifications in Section III. For transversally non-homogeneous materials the structure can be approximated by many layers. However such approach could be computationally expensive as it requires calculation of modified Bessel or exponential functions (of complex argument) for each layer.
A fully anisotropic case is treated in Section IV. The field-matching technique does not work in this case. We consider several possible analytical formulations and choose one with non-singular differential equations. For this choice we describe a simple finite-difference scheme. With a proper permutation of mesh indexes we reduce a sparse matrix with 7 bands to a pentadiagonal one. It allows a fast algorithm of complexity O(N) (N-number of mesh steps) for calculation of impedances for non-homogeneous anisotropic materials. Open boundary conditions are formulated for the case when the last layer has an uniaxial anisotropy in the beam direction.
The finite difference method allows treating of full anisotropy but could be timeconsuming. For the case when the anisotropic layers are thin we suggest in Section V a combination of the field matching and the finite-difference approaches.
Finally in Section VI the described methods are cross-checked on several numerical examples. The algorithms are implemented in numerical code ECHO [11].
We consider a point-charge q moving with constant velocity v through a structure with round or rectangular cross-section. In the following we call the structure "round" if it is axially symmetric. If the structure has a constant width between two perfectly conducting planes and has rectangular cross-sections then we call such structure "rectangular". Fig. 1 shows examples of round and rectangular structures.
In the following we consider only an anisotropic materials with diagonal material permittivity and permeability tensors, where the optical axes coincide with coordinate ones. Hence their diagonals are given by complex vectors , µ.
We assume that the charge is moving along a straight line parallel to the longitudinal axis of the system, and we neglect the influence of the wakefields on the charge motion. For round structures we will use cylindrical coordinates r, ϕ, z. The charge density in the frequency domain can be expanded in Fourier series where r 0 , ϕ 0 are coordinates of the point charge q, β = v/c, c is velocity of light in vacuum, and δ m0 = 1 if m = 1, 0 otherwise.
From the linearity of Maxwell's equations the components of the electromagnetic field can be represented by infinite sums: The electric displacement D and the magnetic induction B are defined using com-plex permittivity and permeability diagonal tensors We do not have to assume any particular frequency dependence. In order to include conductivity and other losses in our code ECHO1D we use the following expressions (here we consider as example r-component): where r is the real part of the complex permettivity, µ r is the real part of the complex permeability, and the loss can be introduced with the help of dielectric loss tangent δ r , magnetic loss tangent δ µ r or/and with AC conductivity following the Drude model [12], where κ r is the DC conductivity of the material and τ r its relaxation time. We use similar expressions for ϕand z-components of the permittivity and the permeability tensors.
For each mode number m we can write an independent system of equations We have reduced the initial three-dimensional problem to an infinite set of independent dimensional problems, Eqs.
(3), for the Fourier componets of the field.
In rectangular case we choose a coordinate system with y in the vertical and x in the horizontal directions; the z coordinate is directed along the beam direction. The structures considered in this paper have constant width 2w in x-direction between two perfectly conducting side walls.
The charge density can be expanded in Fourier series where x 0 , y 0 are coordinates of the point charge. Again it follows from the linearity of Maxwell's equations that the components of electromagnetic field can be represented by infinite sums: For each mode number m we can write an independent system of equations We are interested in coupling impedances as defined in [5,14]. For round pipe the coupling impedance can be written as Z (r 0 , ϕ 0 , r, ϕ, k, γ) = ∞ m=0 Z m (k, γ)I m kr 0 γβ I m kr γβ cos(m(ϕ − ϕ 0 )) + Z sc (r 0 , ϕ 0 , r, ϕ, k, γ), where γ is the relative relativistic energy and we have written explicitly the space For a rectangular pipe the impedance reads where In Eqs.(5, 6) the infinite sum defines a so-called wall impedance. The longitudinal and the transverse wall impedances are connected by Panofsky-Wentzel theorem (see [5] for a detailed discussion).
The wake field effect in time domain is described by a longitudinal wake function which can be obtained by the Fourier transform of the longitudinal impedance where s is the distance between the source and the test particles [14].

III. FIELD MATCHING FOR UNIAXIAL ANISOTROPY
In the general anisotropic case from system of first-order Eqs.
(2) we obtain the second-order coupled equations for z-components of the electric and the magnetic fields: 1 r ∂ ∂r The field matching technique for round and flat isotropic pipes was considered, for example, in [5][6][7][8][9][10]. For the case of uniaxial anisotropy along z-axis we use the same technique, which we describe shortly in this Section.
We consider the uniaxial anisotropy when the permittivity and the permeability tensors are diagonal and for their elements the following relations hold Inside of each layer where the complex permeability and permittivity are constants (independent from r) Eqs.(7) reduce to the decoupled equations A general solution of homogeneous hyperbolic Eqs. (8) can be written in form where I m , K m are modified Bessel functions of complex argument.
In the following we will numerate the layers by index j and r = a j defines interface between the layers with numbers j and j + 1. In order to find the constants C m, j I , C m, j K , D m, j I , D m, j K in Eqs. (9) we can use 4 conditions at the interfaces between the layers: where the radial field components are defined through the longitudinal ones as From Eqs. (9)-(11) at each interface r = a j we obtain the relations where M j is a complex matrix of order 4. We do not write the explicit form of the elements of the matrix M j . They can be written as a combination of modified Bessel functions and the expressions are similar to those obtained in [5] for an isotropic case.
The matrix connecting the coefficients from vacuum layer to the coefficients of the last layer can be found as a matrix product After numerically solving of Eq. (12) the modal longitudinal impedance in Eq. (5) can be found as If the last layer, j = N, is closed with perfectly electric conducting (PEC) material at r = a N , then we use a modified matrix N is a matrix converting the field coefficients in the field components H r , H ϕ and their derivatives: Again  , k µ y,m = k 2 x,m + ν 2 y µ z µ y , ν 2 y = k 2 β −2 − ω 2 2 y µ 2 y .
In the following we consider only the case where the rectangular structure is symmetric in the y-direction (up-bottom symmetry). In this case Eq. (6) has a simpler form Z m (y 0 , y, k, γ) = Z cc m (k, γ) cosh(k y,m y 0 ) cosh(k y,m y) + Z ss m (k, γ) sinh(k y,m y 0 ) sinh(k y,m y). Hence we are looking for the solution of the following system After numerical solution of Eq. (13) the item Z cc m (k, γ) can be found as The item Z ss m (k, γ) can be found from the solution of another problem in the half of the domain with electric boundary condition at the symmetry plane E z,m (0) = 0. We are looking for the solution of the following system After numerical solution of Eq.(14) the item Z ss m (k, γ) can be found as .
If the last layer, j = N, is closed with perfectly conducting material at y = a N then we use a modified matrix in the same way as described above for the round geometry. We will not consider here a rectangular structure without symmetry. In general, matrix M is a composition of matrices for all layers. It can be found and treated in the same way as described in [5] for an isotropic case.

IV. FINITE-DIFFERENCE METHOD FOR FULL ANISOTROPY
In this section we describe a finite-difference method to treat the round and the rectangular structures with arbitrary anisotropic materials. We start with the round case. At the beginning we have to decide which equations to use. The system In order to remove the discontinuity of the azimuthal component in the charge location r 0 we present the azimuthal component of the magnetic field in the form where θ(r) is Heaviside function and H 0 ϕ presents a monopole harmonic of the self field of relativistic charge in free space. Let us note that H s ϕ,m has the meaning of the scattered field only for the lowest monopole mode, m = 0, and the relativistic charge. Another choice could be to take H 0 ϕ,m as a true m-harmonic of the selffield but this introduces additional terms into the right-hand side of Eqs. (15), (16) without any clear improvement of the accuracy of the numerical solution. We introduce one dimensional mesh with shifted positions of the transverse magnetic filed components as shown in Fig. 3. The mesh in material is not equidistant in general. It is chosen to sample the wave length in the material properly and depends on the wavenumber k = ω/c. We use the standard second order approximations of the derivatives [15] and the finite-difference scheme reads 1 (17) where we have introduced the discrete field components h ϕ,i = H s ϕ,m (r i )r i , h r,i+0.5 = µ r (r i+0.5 )H r,m (r i+0.5 )r i+0.5 and the following notation At the axis of axially symmetric geometry we have magnetic boundary condition [H ϕ,r r]| r=0 = 0, ∂ ∂r [µ r H r,m r]| r=0 = 0, and the equations for i = 1 can be written in the form 1 If the exterior boundary is perfectly conducting at r N+0.5 = b then we have electric boundary condition for the magnetic field where the matrix M has dimensions 2N × 2N and the seven band structure shown in Fig. 4 on the left side. In order to use a direct method of solution of linear system (18) we introduce the permutation matrix P σ defined by permutation of indexis It converts the sparse seven band matrix M in pentadiagonal form P σ MP T σ shown in Fig. 4 on the right side. , G = ikZ 0 2π(γ 2 − 1) .
In the case of rectangular geometry we again consider only the case with symmetry plane at y = 0. In this case we have to solve two problems in half of the computational domain. The first problem for Z cc has a magnetic boudary condition at the symmetry plane (H z,m (0) = 0) and we approximate it in the same way as it was done at the axis for round geometry. The second problem for Z ss has an electric boundary condition (E z,m = 0) at the symmetry plane and we approximate it in the same way as it was done for round geometry at PEC boundary.
The longitudinal electric field component and the impedance in the rectangular case can be found as in analytical form, Eq.(9), the middle layer is anisotropic and could be treated only with finite-difference method. Let us denote the coefficients in the first layer as C m,1 I , C m,1 K , D m,1 I , D m,1 K and the coefficients in the third layer as C m,3 I , C m,3 K , D m,3 I , D m,3 K . In order to use the matrix approach of Section III we need to find matrix M FD 13 , converting the first set of coefficients in the second one: The matrix M 13 can be found as a product of several simple complex matrices of size 4 × 4: where M CF h r = (h ϕ,2 , h ϕ,3 , ..., h ϕ,N+1 , h r,1.5 , h r,2.5 , ..., h r,N+0.5 ) t , The matrix M r of system (24) has the form shown in Fig. 6   The field components Analogously we will find the third and the fourth columns of this matrix.
As can be seen from the above description we need to solve the problem 4 times in the anisotropic layer only. If the layer is thin then the suggested method is faster than the finite-difference method of the previous section where the whole domain has to be discretized to sample the electromagnetic field everywhere. At the rectangular geometry the algorithm is exactly the same with corresponding equations for the rectangular case.

VI. NUMERICAL EXAMPLES
Recently, experimental demonstration of energy modulations in dielectric pipes was observed at the PITZ facility [17]. The experiment was performed with a dielectric pipe with an isotropic dielectric layer of permittivity = 4.41 0 . The layer starts at radius a 0 = 0.45 mm and is closed with PEC at a 1 = 0.55 mm. We take this dielectric pipe as our first example and calculate the steady-state wake of a relativistic Gaussian bunch with rms length σ z = 25 µm. In Fig. 7 we show the longitudinal and the transverse wake potentials near the pipe axis. The longitudinal wake potential for the charge distribution λ(s) is defined as The transverse wake potential is defined analogously and W ⊥ (s) means here the dipole component of the transverse wake normalized by offset [14]. ECHO2D [11]. In order to obtain the steady-state wake in time-domain we have subtracted the wake for pipe of length 10 cm from the wake of pipe of length 11 cm.
The agreement of the curves from two different methods confirms the correctness of the results. In Fig. 8 Table I. Let us note that in this example we have used a small conductivity κ = 1 S/m to resolve the real part of the impedance.
For the same aperture size the cylindrical geometry allows to obtain the highest accelerating gradients. Due to technological difficulties in preparing cylindrical structures with stringent requirements to tolerances the rectangular structures are

Method Round Rectangular
Field Matching (Section III) 31 5 Finite-Difference (Section IV)  MeV, charge 100 nC and bunch length σ z = 1.5mm is considered. The dependence of the longitudinal electric field component E z at the symmetry axis produced by the bunch on the distance s = vt − z behind it is shown in Fig. 9. The solid line corresponds to anisotropic sapphire, the dashed line corresponds to isotropic filling. The wave number k was sampled from 1 m −1 to 20e4 m −1 with step 0.2 and we have calculated 5 the lowest odd Fourier harmonics in Eq. (6). At this example we used a small conductivity κ = 0.05 S/m to resolve the real part of the impedance.
The data in Fig. 9 agree with the results published in [3]. A frequency shift with a little influence on the wake field amplitudes can be seen.
The execution times of different methods discussed in this paper for the rectangular example are shown in Table I. It can be seen again that for the same accuracy the combined method requires less computational time as compared to a fully finitedifference one.