Fields and Characteristic Impedances of Dipole and Quadrupole Cylindrical Stripline Kickers

We present semi-analytical methods for calculating the electromagnetic field in dipole and quadrupole stripline kickers with curved plates of infinitesimal thickness. Two different methods are used to solve Laplace’s equation by reducing it either to a single or to two coupled matrix equations; they are shown to yield equivalent results. The kickers plates define a set of coupled transmission lines and the characteristic impedances of modes relevant to each configuration are calculated. The solutions are compared with those obtained from a finite element solver (FEMM) and found to be in good agreement. Mode matching to an external impedance determines the kicker geometry and this is discussed for both kicker types. We show that a heuristic scaling law can be used to determine the dependence of the characteristic impedance on plate thickness. The solutions found by semi-analytical methods can be used as a starting point for a more detailed kicker design.


Introduction
Transverse dipole kickers change the transverse momentum of a beam in an accelerator and have multiple applications, e.g. in systems for injection and extraction, feedback, tune measurement etc [1,2]. Transverse quadrupole kickers are used in exciting coherent quadrupole oscillations in space charge dominated beams. Stripline kickers (dipole and quadrupole) are often preferred because of their relative simplicity and fast response time. An application of interest where both types of kickers are needed is the generation of beam echoes [3,4,5,6,7]. Detailed designs are usually done with electromagnetic codes which solve for the fields using a variety of numerical schemes, see e.g. [8,9]. In this paper, we focus instead on analytical methods to solve Laplace's equation. After incorporating the boundary conditions, the solution is expressed as a series whose coefficients are obtained from matrix equation(s) of infinite dimension. The latter are then truncated and solved numerically. This approach leads to general insights about how the fields and characteristic impedances depend on kicker geometry. This study was motivated by the need of these kickers for generating beam echoes in the IOTA ring at Fermilab [10]. The small size of this ring (40 m circumference) calls for compact kickers. Therefore, a design objective is to maximize the electric field (dipole) or field gradient (quadrupole) subject to the constraint of proper matching to external loads. A few simplifying assumptions are made in the analysis, an important one being that the electrode plates are of infinitesimal thickness. We also assume circular symmetry so the electrodes are arc shaped. The two analytical methods presented here were originally applied to the design of striplines for microwave devices [11]; one of them was later used in the analysis of a pickup with a single stripline [12]. In Section 2 we introduce both methods and use them to analyze a dipole kicker; the quadrupole kicker is discussed in Section 3. We compare our semi-analytical results with those from a finite element based code (FEMM) in Section 4. In Section 5 we consider the influence of plate thickness and derive a heuristic scaling law. Our conclusions are presented in Section 6.

Dipole Kicker with circular symmetry
In this section and the following two others, we make the following assumptions: a) the electrode plates are arc shaped and infinitesimally thin, b) the electrode plates and beam pipe are perfect conductors and c) all plates have exactly the same shape and coverage angle, and they are placed at exactly symmetric locations inside the beam pipe. We first consider the two plate dipole kicker configuration. There are two modes to consider: the operational mode or odd mode where the plates are at equal and opposite voltages resulting -to lowest order-in a dipole field and the even mode where both plates are at the same voltage. The even mode is relevant because it is excited by the circulating beam which, assuming it is centered, induces identical charges (a fraction of the beam charge) and voltages on all plates. A more complete discussion of the odd and even modes can be found for example in [13]. If the beam current is high enough for beam instabilities to become a concern, the characteristic impedance of the even mode should be optimized to prevent the field in the resonator formed by the plates from acting back on the circulating beam [9]. As is discussed later in Section 2.4, in general matching the geometric means of both modal impedances to that of the external lines may be the best compromise.

Potential for the Odd Mode
We consider two arc shaped electrodes held at a constant voltage ±V p , a schematic is shown in Fig. 1. The rods supporting the plates are omitted in this sketch. For typical external voltages, the relative contribution of the beam induced voltage to the total voltage is negli-gible so the potential can be assumed to obey Laplace's equation. In two dimensional polar coordinates, (r, θ ) we have For a well-posed problem, the specified boundary conditions ensure a unique solution. As- The plates are located at a radius b, and the beam pipe at a radius a. Each plate subtends an angle of magnitude 2θ 0 . We assume the left and right plates are respectively at voltages V p and −V p so that a positively charged particle is kicked in the positive x direction i.e. to the right. Thus Φ(r = b, θ ) = −V p on plate P 1 which extends over the angles −θ 0 ≤ θ ≤ θ 0 , and Φ(r = b, θ ) = V p over plate P 2 : π − θ 0 ≤ π + θ 0 . The potential must be continuous across the entire boundary r = b. In the region exterior to the plates, the potential vanishes on the beampipe Φ(a, θ ) = 0. Across the interface between the interior and exterior regions (r = b), the radial component of the electric field is continuous on the intervals where no electrode is present: where the gaps G 1 , G 2 have the domains: G 1 : θ 0 ≤ θ ≤ π − θ 0 and G 2 : π + θ 0 ≤ θ ≤ 2π −θ 0 respectively. Due to the presence of charge on the electrode surfaces, it is necessary to consider the interior and exterior regions separately.
The potential within the interior of the plates must be be well behaved as r → 0. This eliminates the coefficients a 0 , b m .
where a m has been absorbed into the redefined coefficients c m , d m . The potential is symmetric about the x axis or Φ(b, 2π − θ ) = Φ(b, θ ), so d m = 0. Furthermore, the anti-symmetry of the potential with respect to the vertical y axis implies Φ(b, π − θ ) = −Φ(b, θ ). Imposing this requirement in Eq.(2.4), one concludes that c 0 = 0 and m = odd.
For the potential exterior to the plates we start from the general form where the coefficients A m , B m ,C m , D m are different from the coefficients a m , b m , c m , d m of the interior solution. They are determined from the boundary conditions on the exterior potential. Requiring that this potential vanish at r = a implies c 0 = −A 0 ln a, B m = −A m a 2m .
Matching the interior and exterior solutions at r = b yields A 0 = 0 = D m , the index m = 1, 3, 5 . . ., and the coefficients A m can be absorbed into C m which can be expressed in terms of the interior coefficients c m as C m [1 − (a/b) 2m ] = c m where m is odd. Expressed in terms of the interior coefficients, the solutions for the interior and exterior potentials are where we have introduced new scaled dimensionless coefficients X m = b m c m /V p . Imposing the boundary condition on the right plate at r = b and matching the normal derivative across the gaps at r = b yields respectively ∑ m=1,3,...
where the dimensionless geometric coefficient g m is defined as We note that g m decreases with increasing index and approaches 1 as m → ∞.
Integrating Eq. (2.6) over the angular extent of the plate at r = b and Eq.(2.9) over half of the top gap G 1 : θ 0 ≤ θ ≤ π/2 (the integral over the complete gap vanishes because of antisymmetry) yields the two equations ∑ m=1,3,...
These are integral conditions which must be satisfied over the plates and the gaps respectively. The coefficients X m must however be found from the local conditions in Eq,(2.8) and Eq.(2.9) which are valid at every point within their respective domains. Below we discuss two methods for determining them.

Least Squares Method
We follow the method used in [11] which parallels a development by Sommerfeld in [14] to treat the problem of light waves reflecting off a curved mirror. Essentially, the method consists of determining the expansion coefficients so as to minimize the quadratic residual error on the boundary conditions. Taking the sum of the squared difference of Eqs. (2.8) and (2.9) and integrating over the appropriate azimuthal ranges yields an error function as (2.13) The minimum residual is obtained by setting the partial derivatives to zero ∂ Err(X) ∂ X j = 0 which yields matrix equations for the coefficients. Define the vector b with components b n and matrix A with elements A mn as follows b n = θ 0 −θ 0 cos nθ dθ = 2 n sin nθ 0 (2.14) The diagonal elements follow from the off-diagonal elements on using lim m→n sin(n − m)θ 0 /(n − m) = θ 0 . This matrix A is symmetric, A mn = A nm . These elements will arise in all the situations to be discussed in this paper for both the dipole and quadrupole kickers.
Collecting terms leads to the matrix equation B · X = −b or in component form The matrix B is a non-singular, square, symmetric matrix of dimension N × N to solve for the N coefficients X 1 , X 3 , . . . , X 2N−1 . This matrix equation (2.17) must be truncated and solved numerically.

Projection Method
This follows the second method investigated in [11] which is referred to as the "simple integration" method. We choose to call it the projection method since it is based on projecting the coefficients X n on to a basis set of harmonic functions. Multiplying Eq.(2.8) by cos nθ and integrating over the plate on the right P 1 : −θ 0 ≤ θ ≤ θ 0 leads to the set of equations ∑ m=odd A mn X m + b n = 0, n odd (2.20) These coefficients form a potential that only satisfies the boundary conditions on the plates but not in the gaps. The matrix A is also singular which is a consequence of the fact that it does not specify a unique potential.
Multiplying Eq.(2.9) by cos nθ and integrating over the gap from θ 0 to π − θ 0 yields the set of equations C nn = [1 + ng n (a, b)]A nn − n π 2 g n (a, b), n = odd (2.23) The matrix C is in general non-singular and can be used to numerically find the desired coefficients X m .

Potential for the Even Mode
Here we consider the potential when both plates are at the same voltage. This mode is excited when the beam induces image currents and voltage difference with respect to the beampipe. This is the so-called even mode or common mode and its characteristic impedance is involved in matching to the external impedances, as is discussed later in Section 2.4.
We assume that both plates are at a positive voltage V b . Now the potential is symmetric about the y axis (as opposed to the antisymmetry in the odd mode) and has identical symmetry about the x axis, i.e. Φ(b, π − θ ) = Φ(b, θ ), Φ(b, 2π − θ ) = Φ(b, θ ). We start with the form for the interior potential in Eq. (2.4). Symmetry about the x axis requires that d m = 0, while symmetry about the y axis requires m is even. For the exterior solution we start with the general form in Eq. (2.5). Requiring that the external potential vanishes at r = a and matching the exterior and interior potentials at r = b ∀θ yields its form. We introduce the scaled dimensionless coefficients X 0 = c 0 /V b , X m = b m c m /V b . The two even mode potentials can be written as Matching the two potentials on the plates and their radial derivatives in the gaps leads to the boundary conditions The integral conditions obtained by integrating Eq. For the sake of brevity, we consider only the projection method to determine the potential in this mode. Proceeding as before, i.e. multiplying Eq. (2.27) and Eq. (2.28) by cos nθ and integrating over the appropriate range of θ , we obtain b n = X 0 b n + ∑ m=even, =n X m A m,n + X n A n,n n = even X m mg m A m,n + X n ng n (π/2 − A n,n ) n = even These equations can be combined to yield the matrix system Once the X n , n > 0 are found, X 0 can be found from either of the integral conditions in Eq.

Electric and Magnetic fields in the odd mode
From the interior potential, it follows that the electric fields in the interior along the Cartesian axes acting on a particle with polar coordinates (r < b, θ ) are We expect (and verify in Section 4 ) that the coefficient magnitudes |X m | decrease with increasing order. Along the x axis, the horizontal field E x has its maximum value while E y vanishes along both the horizontal and vertical axes. The first term in E y has a maximum at θ = π/4, the second term along π/8 etc. Thus for small enough beam size (σ ⊥ b, σ ⊥ is the transverse rms size) the field in this kicker approaches that of a linear dipole magnet, while for larger beam sizes the beam experiences nonlinear kicks in both directions.
The electric E and magnetic B fields in a TEM wave propagating along +ẑ in a transmission line obeyẑ A particle with charge q propagating along −ẑ or in a direction opposite to that of the EM wave, will experience a force with horizontal component The change in momentum ∆p x due to this force from a kicker of length L k is found from ∆p x = L k β c F x while the angular kick ∆x is given by ∆p x = m 0 γβ c∆x , where m 0 is the rest mass. Hence the total horizontal angular kick is which is the sum of kicks from the electric and magnetic fields. If the particle propagates in the same direction as the wave, the two forces oppose each other leading to a near cancellation for relativistic particles. At low energies, for example the 2 MeV proton beam in IOTA has β = 0.07, the magnetic kick is a small fraction of the kick from the electric field, so the relative direction of propagation of the wave and particles does not matter much. We can approximately calculate the dipole kick by assuming X 1 = −1 and all X m = 0, m > 1. Assuming a plate voltage of 1 kV, a plate radius of 20 mm, a compact kicker length of 20 cm, the kick on the IOTA beam is ∆x = 2.1 mrad. In terms of the average beam size at a location with β x = β av = 1.2 m and average beam size σ av = 2.2 mm, this amounts to a kick β x ∆x 1.2σ av . If instead we apply the above estimate to the existing injection stripline kicker [17] where V p = 25kV, L k = 0.635m, b = 0.02m, we have a beam kick β x ∆x 92σ av . This is considerably larger than required (∼ 5 − 10σ x ) in order to explore the nonlinear aspects of the dipole kick on echoes [7].

Characteristic Impedance
An arrangement of n deflecting plates enclosed by a conducting beam pipe forms a set of n coupled transmission lines. For a TEM wave, in the frequency domain the voltage and current amplitudes V i and I i associated with each one of the plates are locally related to each other through the two relations where V and I are vectors of dimension n while k, ω are the spatial and angular frequencies and L , C are the distributed inductance and capacitance matrices. The equations express the fact that current in any one of the conductors induces a proportional voltage in all the others and vice-versa. Combining both equations yields the dispersion relation where we used the fact that the wave velocity c = ω/k and I unit is the unit diagonal matrix. Note that by reciprocity, the matrices L , C as well as their product are symmetric for any arrangement of the plates (symmetric or not). Using a similarity transformation, equation 2.39 can be put in a form where the inductance matrix is diagonal. The required transformation is provided by the matrix U whose columns are the eigenvectors of L .
is known as the characteristic impedance matrix.
For an n-fold symmetric plate arrangement, the number of independent entries of L (or C ) is reduced and all the the self-impedances L ii = L 11 while the off diagonal L i j depend only on the the angular distance between the electrodes i and j. For a dipole kicker, n = 2 and one can verify that the eigenvectors are u o = (1, −1) and u e = (1, 1). These eigenvectors, which are shared by the capacitance and characteristic impedance matrices, define the so-called coupled modes. Any arbitrary excitation can be expressed as a linear combination of the latter. Using the dispersion relation, the eigenvalues of the characteristic impedance matrix Z c,odd and Z c,even may be expressed in terms of those of the L and C matrices to define the effective capacitance and inductance of the even and odd modes C e , L e ,C o , L o .
It follows that in general Z c,even ≥ Z c,odd . When each stripline is terminated by an impedance equal to the characteristic impedance of a given mode, there is no reflection of that mode.
Assuming that all terminations have impedance Z L , both modes are perfectly matched when Z c,even = Z c,odd = Z L . However, this condition is too restrictive since it implies C 12 /C 11 → 0 which happens with increasing plate separation or b/a → 1. For a kicker, the following weaker requirements are sufficient: (1) no power injected in odd (or difference) mode is reflected back to the generator (2) power deposited in the even (or common) mode is coupled out of the striplines. (1) and (2) are satisfied without matching either the even or odd modes separately, but instead when is fulfilled. In fact, when 2.44 holds, the plate arrangement is a directional coupler. However, for high current applications as mentioned previously, it may be best to match the even mode to Z L .
We now calculate the frequency independent (low frequency) part of the characteristic impedance. By definition, a mode characteristic impedance is the ratio of its voltage V p and current I p mode amplitudes : Z c = V p /I p . The current I p can be expressed in terms of the surface current density, i.e. the current per unit length normal to the direction of current flow. Let K p be the current density on a plate where dl is an element of length and L p defines the contour of the plate. At the interface between two media (vacuum in our case), the discontinuity between the tangential components of the magnetic field on either side of the interface is given by [15] wheren is the unit normal from media 1 (region interior to the plates) towards media 2 (region exterior to the plates). In the second equality we have assumed the media are linear so that B = µ H, µ is the magnetic permeability. The normal component of the B field is continuous across the plate. For a TEM wave propagating in the stripline, the E and B fields are orthogonal everywhere to the direction of propagation. We have c B =ẑ × E, use the relationn ×ẑ × E = (n · E)ẑ and let µ 1 = µ 2 = µ 0 (vacuum permeability) to write the surface current density on the plate in terms of the discontinuity in the normal (or radial) components of the electric field across the plate.
Next we calculate the characteristic impedance of the odd and even modes. The transverse and longitudinal beam coupling impedances are proportional to the characteristic impedances of the odd and even modes respectively [2,16].

Odd Mode Characteristic Impedance
Using the potential forms in Eqs.(2.6) and (2.7), we have for the surface current density The current on either plate is Hence the characteristic impedance is where we used the integral condition in Eq.(2.12) in the second equality above. Eq. (2.50) shows for example that Z c,odd is determined entirely by the coverage angle θ 0 and the ratio b/a and not by the specific values of a, b. As the ratio b/a increases, Z c,odd decreases and Z c,odd → 0 when b/a → 1.

Even Mode Characteristic Impedance
The surface current density defined in terms of the discontinuity in the radial electric field across a plate is The current on a plate is where in the last step we used the integral condition in Eq.(2.30 ). Hence the characteristic impedance of the even mode is This expression resembles the characteristic impedance of a coaxial line with a single cable, Z c = Z 0 ln(a/b)/(2π) and differs by a factor of two from a similar expression for the characteristic impedance of a single stripwire kicker [12].

Quadrupole Kicker with circular symmetry
For a four-fold symmetric quadrupole kicker, the inductance (capacitance) matrix has three independent entries and the four characteristic impedances are Modes 1 and 2 are known as dipole modes, mode 3 will be referred to as the sum mode (a beam induced mode where all plates are at the same potential) while mode 4 as the quadrupole mode (the mode which applies a quadrupolar kick). From the above definitions, it follows that in general Z c,sum ≡ Z c3 ≥ Z c,quad ≡ Z c4 . In analogy with the dipole kicker case, it can be shown that with individual lines terminated with a load Z L , an incident voltage wave sent on any one of the lines will not produce any reflection provided that the condition is fulfilled. In this section we again solve for the potential and determine the quadrupole and sum modes characteristic impedances following the method of the previous section.

Potential solution for the quadrupole mode
The plates are numbered in anti-clockwise order starting from the right, a sketch is shown in Fig. 2. In this configuration, the potentials on the plates alternate in sign on adjacent plates with Φ(r = b, θ ) = −V p on plate P 1 on the right which extends over the angles −θ 0 ≤ θ ≤ θ 0 , Φ(r = b, θ ) = V p over plate P 2 : π/2 − θ 0 ≤ π/2 + θ 0 etc. The gaps G 1 , G 2 , G 3 , G 4 extend over the angles not covered by the plates, e.g. G 1 : θ 0 ≤ θ ≤ π/2 − θ 0 , G 2 : π/2 + θ 0 ≤ θ ≤ π − θ 0 etc. The general expressions for the potentials Φ in (r, θ ) interior and Φ ex (r, θ ) external to the plates are respectively, implies that m = even. Since the plates are symmetric, the potential is anti-symmetric about the lines midway between adjacent plates (at opposite voltages) along θ = π/4, 3π/4. These anti-symmetries can be written θ ). These imply c 0 = 0  We now write down the matrix equations for the two methods discussed previously. The error function to minimize in the least squares method has the same form as in Eq.(2.13), except that the second integral (over the gap) runs from θ 0 to π/2 − θ 0 and the index m runs over the values 2, 6, 10, . . . . Minimizing the error function yields the matrix equation B · X = −b where the the off diagonal elements of the matrixB are the same as for the matrix B for the odd mode in the dipole case while the diagonal elements arē B nn = A n,n + n 2 g n (a, b) 2 π 4 − A n,n , n = 2, 6, 10, . . .

(3.8)
The π/2 term in B nn replaced by π/4 inB nn and the indices have different values.
To apply the projection method, multiplying the first boundary condition by cos nθ and integrating over a plate leads to exactly the same as Eq. Hence the matrix equation isC · X = −b whereC is similarly relates to the matrix C for the dipole odd mode defined in Eq. (2.23) and (2.24) as the matricesB and B above.

Potential for the sum mode
In this mode, all plates are at the same voltage. Now, we have symmetry about the axes at ±45 • in addition to the symmetries about the (x, y) axes. As with the quadrupole mode, the symmetries about the x, y axes lead to d m = 0, m = even The symmetries about the other axes along the ±45 • angles imply m = 4, 8, ... Hence, the potential in the interior and exterior can be written as These are of the same form as for the even mode in the dipole kicker, but for for the indices.

Electric and Magnetic fields, Characteristic Impedance
We consider first the fields in the quadrupole mode. The electric fields in Cartesian coordinates are Keeping only the first term gives us the quadrupole fields Using the expressions for the forces derived above, we have for the quadrupole kicks from a kicker of length L k Hence the integrated quadrupole gradient or inverse focal length defined from ∆x = −K q x is We now estimate the quadrupole kick for the 2 MeV IOTA proton beam referred to in Section 2. We assume a plate voltage V p = 1 kV, the kicker length to be L k = 0.2 m, the plate radius b = 0.02 m and use the approximation X 2 = 1, X m = 0, m > 2. This yields K q = 0.21 m −1 , or the dimensionless quadrupole strength q = β x K q = 0.26. According to the theory of nonlinear echoes [7], this value of the quadrupole kicker strength q will suffice for nonlinear effects of the quadrupoles on the echoes to be observable.
The characteristic impedance can be calculated similarly as for the dipole kicker. We have for the quadrupole mode, For the sum mode, the characteristic impedance is As discussed earlier, the geometric mean of the quadrupole mode impedance and the sum mode impedance is matched to that of the external load Z L , or This seems to be the adopted matching scheme in some quadrupole kicker designs [18,19].

Numerical solutions and comparisons with FEMM
In this section we compare the fields found using the two methods described in Sections 2.1.1, 2.1.2 as well as those obtained from the 2D electrostatic (and magnetostatic) code FEMM. This program uses the finite element method to solve for the potential and is available at [20]. The domain of interest is subdivided into triangular regions; the program uses quadratic Lagrangian interpolation over these regions and solves for the potential at the nodal locations.
The Fourier series converges poorly near the tips of the plates due to discontinuities in the derivative of the potential (Gibbs phenomena). A Gaussian filter is applied to improve convergence by smoothing them out. After filtering, the series ∑ m X m cos mθ becomes ∑ m exp[−m 2 /(2σ 2 )]X m cos mθ where σ is the smoothing parameter (we used σ = 100/(2π)). For both methods, we found that after 100 terms in the matrix equations the solutions change by less than 1% with the addition of more terms. The convergence rate with both methods is about the same, the rate with the projection method is marginally faster. We used 200 terms in the results discussed here.

Potential and fields in a dipole kicker
In this section we first compare the potential obtained with two series expansions to the FEMM results. Next we discuss some insights they provide on the influence of the geometrical parameters. Figure 3 compares the potential solutions obtained using the two series expansions with the FEMM results. The potential (scaled by V p ) is shown as a function of θ for three values of r ≤ b with the kicker's geometric parameters fixed at a = 25 mm, b = 20 mm and θ 0 = π/3. Both series solutions yield nearly identical results. At the plate radius r = b, both methods lead to the scaled potential equal to -1 and +1 on the right and the left plates respectively. The plots on the right in Fig.3 show the absolute difference with FEMM results: ≤ 5% everywhere except near the tips of the plates where it is ∼20%. This is likely due to a combination of two factors: (1) the filtering applied to the analytic solutions increases the difference by ∼ 5% at the tips and (2) the FEMM polynomial basis functions can not model the singular variation of the potential in the vicinity of a sharp edge without resorting to an extremely dense mesh. In the central region of the beampipe within the area occupied by the beam, the differences are negligible. We verified that the above difference bounds are valid for values of θ 0 in the range 0.2π ≤ θ 0 ≤ 0.45π; this should cover most cases of practical interest.
It is instructive to consider the behavior of the coefficients X m in the Fourier expansion of the potential. Fig. 4 shows the dependence of the two lowest order coefficients (in the odd mode) on the geometric parameters θ 0 and b/a. We observe that X 1 is always negative and its magnitude increases monotonically with θ 0 . When X 1 = −1, all the other coefficients sum to zero and the potential is nearly linear in the x coordinate, resulting in a uniform horizontal electric field. This is true for different values of b/a. These values will need to be checked against the requirement of matching the characteristic impedance, to be discussed later. The top right plot in Fig. 4 shows that X 1 is mostly constant for b/a ≤ 0.6 and then decreases in magnitude for b/a > 0.6 where the change decreases as θ 0 increases. For example, the decrease in X 1 is < 5% for b/a > 0.6. To a good approximation X 1 is independent of b/a for π/3 ≤ θ 0 < π/2. The bottom plot shows that X 3 is an oscillatory function of θ 0 , crossing zero in the range 0.25π ≤ θ 0 < 0.3π for the different b/a. The dependence on b/a is similar to that of X 1 . The behavior of higher order coefficients X m , m > 3 is similar to that of X 3 , with their absolute values decreasing with order m. These lead to the expected conclusion that the potential interior to the plates is mostly independent of the beampipe radius for b/a ≤ 0.6, the presence of the beampipe perturbs this potential significantly only for small coverage angles and as the distance between the plates and the beampipe wall decreases. The left plot in Fig. 5 shows the horizontal electric field E x (0, 0) at the origin as a function of θ 0 for different values of b/a. The important observation here is that while E x (0, 0) initially increases with θ 0 , it eventually saturates around θ 0 0.4π, so further increase in the coverage angle does not increase the electric field by much. The right plot in Fig. 5 shows the horizontal electric field E x (x, 0) along the horizontal axis out to the plates at x = ±20 mm with a = 25 mm. Of the three profiles for different θ 0 , the flattest profile is obtained for θ 0 = 0.25π, which is expected from the above discussion on   For the even mode where the lowest order coefficients are X 0 , X 2 , ..., we observe qualitatively the same behavior as for X 1 , X 3 , ... in the odd mode. The difference is that X 0 is always positive, increases monotonically with θ 0 and reaches a maximum value of +1 as θ 0 → π/2. The higher order coefficients are again oscillatory functions of θ 0 . In comparison to the odd mode case, the coefficients depend more strongly on b/a. We saw previously that the characteristic impedance in either mode is determined entirely by (b/a, θ 0 ). Assuming that Z c,even is matched, we can determine the choice of parameters that result in the desired electric field. Fig. 6 shows the horizontal electric field at the origin as a function of b/a and θ 0 under the constraint that Z c,even = 50Ω. The values of (b/a, θ 0 ) that yield a desired field E x is obtained by taking the intersections of the horizontal line at this E x with the two curves and reading the corresponding values. As an example, the dashed lines show that for a desired E x = 0.06V p /m of the field in the odd mode requires b/a 0.73, θ 0 0.28π.

Potential and fields in a quadrupole kicker
We now compare the quadrupole kicker solutions obtained using the series expansions with FEMM. Once again the least squares method and the projection method give very close results; therefore, only the projection method's results will be discussed. Fig. 7 shows as an example, the potential in the quadrupole mode calculated using the projection method and its difference with the FEMM result. The differences reach 15 -20% at the tips of the plates, depending on the coverage angle θ 0 , but are less than 5% everywhere Figure 7: Left: Potential (using the projection method) in the quadrupole mode as a function of θ for 3 values of r, the largest r = b with θ 0 = π/6 and a = 25 mm, b = 20 mm. Note that for the quadrupole, 0 < θ 0 < π/4, Right: Difference between the projection method and FEMM values for the potential. The difference reaches 15% at the tips of the plates but is less than 5% everywhere else. Figure 8: Scaled horizontal gradient of the electric field ∂ E X /∂ x at the origin as a function of θ 0 (left) and along the x-axis between the plates at x = ±20mm (right). Here b/a = 0.8 else. The dependence of the coefficients X 2 , X 6 , X 10 , .... on θ 0 , b/a in this quadrupole mode mirrors the dependence of X 1 , X 3 , X 5 , ... on these parameters in the dipole odd mode case. Similarly the coefficients X 0 , X 4 , X 8 , ... in the sum mode have a similar dependence on the same parameters as do X 0 , X 2 , X 4 , ... in the dipole even mode case. Fig. 8 shows the gradient of the horizontal electric field scaled by the potential in the quadrupole mode. The left plot shows the gradient at the origin while the right plot shows the gradient along the horizontal axis. The gradient increases with θ 0 but the range over which the gradient stays constant around the origin decreases with increasing θ 0 .

Characteristic impedance
The characteristic impedance of a mode can be found from FEMM using the relationship between Z c and the capacitance per unit length C , namely which was written down in Section 2.4. FEMM is used to calculate the characteristic impedance of a single mode at at time for which this relation is easily derived. in a TEM mode, the electric (E E ) and magnetic (E B ) field energies per unit length are equal. This follows by integrating the volume energy densities (E E , E B ) over the entire cross-sectional area: and where we used c|B| = |E| in a TEM mode. Using E E = (1/2)C V 2 , E B = (1/2)L I 2 (L is the mode inductance per unit length) and E E = E B , we have Z c = V /I = L /C . Using the relation for the wave speed c = 1/ √ L C , Eq.(4.1) follows. FEMM calculates the stored energy E E ; the capacitance and the characteristic impedance are found using the above relations. Fig. 9 shows the characteristic impedance in the even mode of a dipole kicker as a function of the coverage angle θ 0 for different values of b/a. For each value of b/a, three curves are shown. One curve represents the theoretical result in Eq.(2.52), another the result from FEMM and the third is the result from the approximation We observe that the theoretical and FEMM results are in close agreement except at very small θ 0 → 0, a region not of practical interest. We also observe that the approximate expression agrees with the exact results in a small range close to θ 0 = π/2, this range increases with increasing b/a. Fig. 10 shows the allowed values of θ 0 as a function of b/a under the constraints of keeping the characteristic impedance in the odd or the even mode constant at 50 Ω. This plot also shows that for b/a < 0.8, the characteristic impedance is always greater in the even mode while for 0.8 < b/a < 1, the two mode impedances are nearly the same. Choosing b/a in this range and the corresponding θ 0 will allow us to very nearly match both modes to the external impedance. This requires that θ 0 < 0.15π which as can be seen in Fig. 6, lowers the electric field for a given plate voltage. Hence, the same electric field in a dipole kicker with both modes nearly matched requires a higher voltage compared to that in a kicker with only one of the modes matched to the external lines.
Next we discuss the calculation of the quadrupole kicker's characteristic impedances of the two relevant modes. Fig. 11 shows Z c,sum using bosh the projection method and FEMM. As with the dipole kicker, the agreement between the two results is very good except for small values of θ 0 < 0.05π. The behavior of Z c,quad with θ 0 is similar but always obeys Z c,quad ≤ Z c,sum with equality at θ 0 → 0. As functions of b/a, we find that   Z c,quad is nearly independent of b/a while Z c,sum decreases as b/a increases and Z c,quad = Z c,sum → 0 as b/a → 1. Fig. 12 shows the allowed values of θ 0 as a function of b/a when the characteristic impedance of either the quadrupole or sum mode or their geometric mean is kept constant. If the geometric mean is matched to the external load, the black curve in the middle shows the minimum value of (b/a) min ∼ 0.8. This is slightly larger than the minimum b/a for dipoles. We also observe that for b/a 0.85, all three impedances nearly merge so in this range, both modes will be nearly matched assuming the plates have zero thickness.

Characteristic impedance with finite plate thickness
In the analytical treatment we made the assumption that the plates have zero thickness but for a practical device this assumption has to be dropped. In this section we discuss how to extend the above results to plates with finite thickness. Since most of the charge on the plates will move to the edges, we do not expect the thickness to strongly affect the capacitance and therefore the characteristic impedance. However fringe field effects around the plates do have some impact on the field configuration close to the plates. In addition, the curvature of the tips determines the maximum field near the plates. These effects are stronger as the plates get closer to the beampipe (i.e. larger b/a) and also when the number of plates increase causing the fringe fields to overlap.
We consider first the dipole kicker's even mode characteristic impedance. We can argue that the finite thickness reduces the effective radius of the beampipe from a to a value a − t/k f where t is the thickness and k f is a fit factor to be determined. By this argument, the characteristic impedance for the even mode with finite thickness can be obtained from that with zero thickness given by Eq.(2.52) using Z even c,sc (t) = Z c,even (t = 0) For each value of b/a, there is a single value of θ 0 for which the characteristic impedance matches the external load impedance Z L , e.g. see Fig.10 for the zero thickness results. Therefore we find the fit parameter k f for each b/a by minimizing the difference between Z even c,sc (t) and Z L at the value of θ 0 for which Z even c,FEMM (t) = Z L . The fit parameter was determined for several values of b/a ≥ 0.7, since the minimum value of b/a = 0.67 from Fig.  10. This was done for thicknesses in the range 1 ≤ t ≤ 6 mm. Fig. 13 shows the scaled even mode impedance compared with the FEMM values for thickness t = 3, 6 mm and b/a = 0.7, 0.8 using two values of the fit parameter: 1) k f = k min , the exact fit parameter from the minimization and 2) k f = 5.5. We set Z L = 50Ω. While the exact fit parameter varies over the range 4.4 < k min < 6.7, we find that using the approximate value k = 5.5 results in a reasonably good fit (difference |Z c,FEMM − Z c,sc (t, k f = 5.5)| < 1.3Ω) over the range of values 0.7 ≤ b/a ≤ 0.9 and 1 ≤ t ≤ 6 mm. As seen in Fig. 13, the FEMM curve and the scaled curves are close for θ 0 > 0.1π rad, but start to diverge for smaller coverage angles.  We apply the same scaling law with thickness as in Eq,(5.1) for the geometric mean Z c,geom of the two modes of interest in the quadrupoles. Strictly speaking, this scaling should be directly applicable only to the sum mode impedance. Here however, we test the scaling on the impedance Z c,geom that is matched to the external impedance. We found in Section 4 that the allowed range of b/a for zero thickness plates in the quadrupoles at constant Z c,geom = 50Ω is 0.8 ≤ b/a < 1, this range is narrower than in the dipole case. Here we find that over the range 1 ≤ t ≤ 6 mm with b/a = 0.8, the best fit parameter k f ,min varies from 3.6: 3.8 while with b/a = 0.9, k f ,min has a different range 4.9: 5.3. Fig. 14 shows the scaled geometric mean characteristic impedance compared with the FEMM values for b/a = 0.8, 0.9 and t = 3, 6 mm. We observe that for b/a = 0.8, the scaling law applies reasonably well for θ 0 ≥ 0.1π rad. However for b/a = 0.9, the scaling starts to break down especially for the thicker plate. The fact that there is no single value of the fit parameter k f that can be used for different b/a makes this scaling law for the quadrupoles less useful than for the dipoles. Nevertheless for practical purposes, the scaling could be used to determine the θ 0 value for which Z geom = 50Ω even at the extreme values b/a = 0.9,t = 6 mm. This has been confirmed with direct calculations using FEMM. We have verified that similar scaling behavior is observed with the sum mode impedance Z sum except that the fit parameter k f are different.

Conclusions
In this paper we discussed two semi-analytical methods that solve for the potentials, fields and characteristic impedances Z c of the relevant modes in dipole and quadrupole stripline kickers. We assumed that the plates have infinitesimal thickness and the plates and beampipe have circular symmetry. The relevant parameters are (b/a, θ 0 ) where a, b are the beampipe and plate radius respectively and 2θ 0 is the angle spanned by each plate. Reflection symmetries or anti-symmetries as appropriate for the mode, are used and all solutions are expressed in terms of a series of Fourier harmonics, the harmonics depend on the mode and the type of kicker. Two methods are used to find the series coefficients: a least squares method that minimizes the global error on the boundaries and a projection method where the potential is projected onto a set of basis functions. In both cases, one obtains infinite dimensional linear systems (different for each method) which are then truncated and solved numerically. In both cases the series was found to converge to ∼ 1% using the first 100 terms. Comparisons with a finite element code FEMM showed good agreement for both types of kickers. The deviation between the results from the series expansions and FEMM is ≤ 5% everywhere except at the tips of the plates where it can be of the order of 15-20% , depending on the parameters and generally higher in the quadrupole. This is likely due to a combination of the filtering used to damp the Gibbs phenomena in the analytic solutions and inadequate mesh density in FEMM near the tips of the plates. Characteristic impedances for the two modes of interest in each kicker (the odd and even modes in the dipole and the quadrupole and sum modes in the other) were calculated and found also to be in good agreement with those obtained with FEMM. Matching the even mode in the dipole to an external impedance (50 Ω) constrains the allowed values of (b/a, θ 0 ). Fig. 10 shows that this matching requires b/a > 0.67 and the allowed values of θ 0 in this range. A similar plot for quadrupoles is seen in Fig. 12 which shows θ 0 as a function of b/a when either the quadrupole mode, sum mode or the geometric mean of the two modes is matched, In this case, matching the geometric mean requires b/a ≥ 0.8. This lower bound for the quadrupole is more sensitive to the plate thickness (decreases with increasing thickness) than it is for the dipole kicker. To account for the dependence on plate thickness, we tested a heuristic scaling law to obtain the Z c at a finite thickness from the value at zero thickness.
In the case of the dipole kicker, this scaling law with a single value of a fit parameter results in a useful approximation (to within 1 Ω ) to the even mode Z c over a range of thicknesses and b/a values. For the quadrupole, the scaling law does not work with a single value of the fit parameter, but nonetheless can be used to find the correct value of θ 0 , given b/a, thickness and the external impedance.
As mentioned in the introduction, this study was motivated by the need of these kickers for beam echo generation. In this context, kickers are powered for the duration of a single turn; field (or gradient uniformity) is not a primary concern. For other applications where this is an issue, uniformity can be improved by shaping the electrodes. Simulations with FEMM [21] show that for the same applied voltage, straight parallel plates (with comparable dimensions) yield comparable field strengths but with better field (or gradient) uniformity. The solutions presented here can be used to guide the initial design of either dipole or quadrupole stripline kickers before resorting to complex software packages for a detailed design to address other important issues such as minimizing the beam impedances.