Field Stabilization of Alvarez-Type Cavities

Alvarez-type cavities are commonly used to reliably accelerate high quality hadron beams. Optimization of their longitudinal field homogeneity is usually accomplished by post-couplers, i.e. additional rods being integrated into the cavity. This paper instead proposes to use the stems that keep the drift tubes for that purpose. As their individual azimuthal orientations do not change the cavity's undisturbed operational mode, they comprise a set of free parameters that can be used to modify higher mode field patterns. The latter have significant impact on the robustness of the operational mode w.r.t. eventual perturbations. Several optimized stem configurations are presented and benchmarked against each other. The path to obtain these configurations is paved analytically and worked out in detail through simulations. It is shown that the method provides for flat field distributions and very low field tilt sensitivities without insertion of post-couplers.


I. INTRODUCTION
In 1945 L. Alvarez invented rf-cavities to accelerate beams of protons [1]. These cavities became the work horses of many drift tube linacs (DTLs) all over the world [2,3]. They provide a longitudinally constant axial electric and an azimuthally constant magnetic field being referred to as the T M 010 operating mode. Along each cell the radio-frequency (rf)-fields have the same phase implying that the phase advance between adjacent cells is zero. The group velocity v g of a resonating mode is proportional to the slope δω/δk of the dispersion curve. Since the group velocity of the T M 010 mode is zero, there is no power flux along the structure and the field distribution is very sensitive to frequency perturbations of the cells, to power losses, and to beam loading. This problem is particularly serious for long DTLs, where the frequency separation between modes is low [4]. Frequency perturbations usually arise from mechanical errors of the cavity and from thermal distortion during operation which cannot be mitigated completely. For these reasons field stabilization is a major concern in the design and operation of Alvarez-type cavities.
The common solution for field stabilization has been proposed in the 1960ies at the Los Alamos Notional Laboratory [5]. Internal metallic rods extend from the cavity mantle towards the drift tube (DT) without touching them. These post-couplers (PCs) form a resonant structure that couples to the operational mode of the cavity [6]. However, this method does not apply to large tank radii and previous experience with field stabilization of large diameter Alvarez-type cavities shows that stabilization with standard PCs is very difficult or even impossible [7,8].
Alvarez-type cavities with diameters exceeding about one meter need to provide two stems per tube for mechanical robustness. For such cases this paper proposes to use the stems itself for field tuning. A specific tuning method is developed based on dedicated stem configurations to provide for the function of post-couplers in large diameter cavities.
In the following section an equivalent circuit for a cavity equipped with stems is introduced together with 3dfield simulations for an Alvarez-type cavity with constant particle velocity profile. The third section is on dispersion relations from the equivalent circuit and presents the principle of stabilization through orientation of the stems. Afterwards different tuning strategies are established to achieve field flatness optimization and low field tilt sensitivities by simulations during the cavity design stage. In the fifth section perturbations are added to analyse the achievable field tilt sensitivity (TS). The efficiency of the proposed tuning methods is demonstrated through simulations of a cavity with increasing cell length.

II. EQUIVALENT CIRCUIT OF ALVAREZ-TYPE CAVITY
Previous studies showed that the stems play a significant role in the shaping of dispersion curves of T Mmodes [9]. During simulations for the UNILAC poststripper upgrade [10] a considerable influence of the stem configuration on the mode spectrum and on the TS [11] was observed. The equivalent circuit is studied in order to benefit from this effect by introducing a new field stabilization method.
In general, operational Alvarez-type cavities have one or two sets of uniform stems keeping the drift tubes on the axis of the cavity. It is convenient to virtually divide the cavity in an upper part, a lower part, and the drift tubes (Fig. 1). This allows for introduction of three conductor transmission lines being assigned along the tubes, the upper, and the lower part of the tank. This topology is similar to the four conductor line introduced in [12]. The analogue analysis of π−mode structures without stems has been developed and applied to field tuning in [13,14] but tuning with PCs or stems requires more specific analysis. The resulting transmission line model along a length dz on the beam axis being short w.r.t. the tank length is shown in Fig. 2. In this circuit Z u , Z l , and Z t are the longitudinal impedances per length on the upper and lower side of the tank and of the tube array. Y tu , Y tl , and Y ul are the transverse admittances between tube and upper side, tube and lower side, and between upper side and lower side. The parameters are indexed referring to the three parted DTL cross-section introduced above: u (upper part), l (lower part), and t (tube). In the same way double indices are used, e.g. Y tu as transverse admittance between the tube and the upper part of the DTL per length, Y tl as transverse admittance between the tube and lower part of the DTL, and Y ul as transverse admittance between upper and lower part of the DTL. These admittances per length are independent of the length dz. In the further analysis dz will be defined as one cell length βλ comprising two half tubes and one gap implying the assumption that the overall tank length is large compared to the single cell length.

A. Differential Equations of Field Distribution
Applying the sum rule to the equivalent circuit of Fig. 2, the differential voltages along the cavity are given by the longitudinal current through where the change of longitudinal current is the sum of the transverse currents These mesh equations deliver quadratic differential equations for the voltage distribution along the equivalent transmission line with the coefficient matrix Defining and after some substitutions and simplifications one obtains that the system is described by where U t is the voltage along the tube, U tu = U t − U u is the voltage between the drift tube and the upper side of the tank, and U tl = U t − U l is the voltage between the drift tube and the lower side of the tank. The coefficient matrix B is defined as These quadratic differential equations can be solved if the distributions of Z(z) and Y (z) are known. Generally Z(z) and Y (z) are designed to be constant to achieve a constant electric field along the cavity axis. This allows for analytical solutions of the voltage distributions, which deliver different field patterns for different modes.

B. Transverse admittances
Equation (6) can be used to derive the tilt of the electric field E z . This tilt should be kept small for the operation of real cavities which are subject to geometrical imperfections Since Z t is defined by the geometry of the tubes and the tank itself, it must be constant for a given design. The key to minimize the field tilt is minimizing the transverse admittances Y tu and Y tl . The admittances can be varied by the angle between subsequent pairs of stems, the stems density on the upper and lower sides of the tank, the number of stems, and by the diameters of the stems. PCs serve also do adjust the transverse admittance by properly choosing their length [15]. In order to investigate how exactly the stems configuration affects transverse admittances and mode spectra, systematic studies on different stem configurations were conducted. It may be hard to measure or to calculate the values of each inductance and capacitance, not to mention the coupling and mutual inductances among them. A feasible way to determine the values is by deriving the dispersion relations as functions of these unknown values and to conclude these values from few simulated frequencies of T Eand T M -modes.

III. DISPERSION RELATION FOR DIFFERENT STEM CONFIGURATIONS
In order to simplify the analytical treatment but without losing generality, βλ, impedances, and admittances are assumed to be constant along the beam axis. Using the boundary condition that on the front wall (z=0) and on the end wall (z=l) with l as the length of the cavity the quadratic differential equations Equ. (6) are solved by a standing wave. Accordingly, the voltage distribution for mode number n can be expressed as where U t0 , U tu0 , and U tl0 are constant and Equation (11) delivers − n 2 π 2 l 2 as an eigen value for B satisfying where I is the unit matrix. With some transformations the dispersion relation is derived as where impedances Z and admittances Y are implicit functions of the mode frequency ω n . Using this dispersion relation the dependencies between transverse admittance and frequencies of modes can be investigated.
With n = 0 for the basic operational mode ω 0 , this equation simplifies to where one solution is Z u Z t + Z l Z t + Z u Z l = 0 which defines the frequency of the T M 010 -mode and of another T M -mode if Z u ≡ Z l . Equation (14) implies that the operating frequency is independent of the transverse admittances Y , thus confirming that the stem configuration does not change this frequency.
The solutions with Y tu Y ul + Y tl Y ul + Y tu Y tl = 0 correspond to frequencies of two different T E 010 -modes, being modes where just the upper part or just the lower part of the cavity resonates. Both T E 010 modes are independent of the cavity's longitudinal tube and gap structure. For a given cavity the resonance frequencies can be determined from 3d CST [16] simulations. Additionally, they can be expressed analytically through the dispersion curve of Equ.  the stem configuration. According to the tube field distribution shown in Fig. 4 the tube impedance is where L t is the inductance of the drift tube and C t is the capacitance of the gap. The expressions of the other elements will be established from the stem configuration of the specific case.

A. Cavity without stems
If there are no stems at all, just the capacitance C between tank mantle and tubes is considered in the transverse direction. The upper side and the lower side of the tank mirror each other and the expressions get FIG. 4. Simulated current distribution on the drift tube surface corresponding to the inductance Lt and electric gap field corresponding to the capacitance Ct between tubes. The longitudinal electric field nearby the tubes is merged into this capacitance as well.
Equation (13) then simplifies to where ω n is the frequency of mode number n. The operating frequency (n = 0) can be expressed analytically as from the equivalent circuit and it is known to be equal to 2π×108.4 MHz. Equation (17) fits the dispersion curve from simulations shown in Fig. 5 to an error of less than 0.1%, if C/C t = 2.57 is assumed. Simulations and Equ. (17) demonstrate that there are no T E-modes being exited in this case as their resonance frequencies are equal to zero.
FIG. 5. Dispersion curves of T M0n0-modes for an Alvareztype cavity without stems from analytic calculation (green) and from simulations (brown).

B. Cross and Alternating V-Configuration
In the following the cross-configuration and the alternating V-configuration as shown in Fig. 6 are treated. The transverse elements of the equivalent circuit are illustrated in Fig. 7 together with the field distribution from simulations. L s is the inductance of stems and C is the capacitance between the tank mantle and the tube together with some fraction of the stem. Using these equivalent elements the explicit expression for inductances and admittances are simplifying Equ. (13) to There are two physical solutions for ω 2 n leading to two different dispersion curves. The first represents T E-modes with the basic mode frequency ω e0 , which follows the equation Y tu (ω e0 ) = Y tl (ω e0 ) = 0. The second dispersion function represents T M -modes and follows Z t (ω 0 ) = 0.
In analogy to the procedure described in the previous subsection one obtains ω e0 = 1 ternating V-configuration C Ct is larger than in the previous case of a cavity without stems as the capacitance C between tank mantle and tubes is increased significantly by the stems. This is seen also from the field map shown in Fig. 7, where the part of the stem being close to the tube also provides some electric field pointing towards the tank mantle thus providing additional capacitance. Figure 8 also shows that with this stem configuration the dispersion curve for T M 01(n>1) -bands has negative slope, the T E-mode frequencies are higher than the T Mmode frequencies, and its dispersion curve has positive slope. This is from the negative traverse admittances Y tu and Y tl . The T M 011 -frequency can be pushed below the T M 010 -frequency. For tuning with PCs such an inversion of the dispersion curve slope happens if more PCs are applied than actually being needed for stabilization.

C. Constant V-configuration
This subsection is on the well-known case that all stems are distributed on the same side of the tank as depicted in Fig. 3. The transverse elements of the equivalent circuit are illustrated in Fig. 9 together with the field distribution from simulations. Although a case of two stems per drift tube is shown here, the presented formalism can be applied to the single stem case as well, since the latter is the special case of an angle of zero degrees between the two stems that bare the same tube. The equiva- lent circuit of an Alvarez-type cavity section with stems mounted at the upper cavity part is shown in Fig. 10.
In the circuit L su , L sl , and L θ are the inductances associated with the flow of longitudinal and transverse currents along the cavity wall per cell length dz. C u is the capacitance between the tube and the upper side of the tank, C l is the capacitance between the tube and the lower side of the tank. Both inhabit components from parts of the stems being close to the tube. L su and L sl are the inductances of the upper and lower side stems. As before in the circuit equations, constant cell length is assumed and the impedances and admittances are functions of the frequency ω. From the circuit the following expressions for longitudinal impedances per length Z and transverse admittances per length Y are derived: Plugging these expressions into Equ. (13) gives a cubic equation for ω 2 n , i.e. three different solutions for each mode number n lead to three different dispersion curves plotted in Fig. 11. The upper dispersion curve corresponds to the resonating of the upper side of tank (Fig. 9a), and the lower curve to the resonating of the lower side of tank (Fig. 9b). The central dispersion curve starting from 108.4 MHz represents T M -modes and its slope is smaller compared to the no stem case according to Equ. (13). Simulation of the constant V-configuration also revealed that if just one pair of stems is rotated by 180 degrees, the slope of the T M 011 -field distribution will be inverted locally. This local inversion of d 2 Ut dz 2 is caused by the locally inverted transverse admittances. Thus one can determine the longitudinal distribution of transverse admittances by analysing the longitudinal field distribution of the T M 011 -mode. This will be exploited in the tuning method introduced in the following section.  Fig. 9.

IV. ANALYTIC FIELD FLATNESS STABILIZATION
The stem configuration determines the shapes of the branches of the dispersion curves, preserving the operational mode. According to the previous analysis, tuning through the stem configuration is by adjusting the transverse admittances to zero along the whole cavity. Tuning by post-couplers aims at closing the stop-bands, which is effectively the same thing but explained in a different way. The scaling of transverse admittances Y tu and Y tl defines the following behaviour of the cavity under tuning: • field tilt sensitivity of the T M 010 -mode: if the cavity operates at nominal frequency (e.g.. 108.4 MHz), the field tilt scales through the transverse admittance according to Equ. (8).
• slope of the dispersion curves of the T M -modes, notably in the interval between T M 010 and T M 011 (from Equ. (19)): if higher T M -mode frequencies ω (n>0) are close to the nominal mode ω 0 , the equation of dispersion curves simplifies to implying that if ω e0 gets close to ω 0 , ω n>0 will be pushed away from ω 0 . An alternative explanation for the relationship between field tilt and frequency separation is that from perturbation theory the field distribution of the T M 010 -mode with perturbations can be expressed through linear superposition of all T M -mode fields along a uniform cavity where U n is the voltage distribution of T M 01nmodes without perturbation, U n = cos( nπz l ), and the factor a n being proportional to 1 ωn−ω0 .
• field distribution of all kinds of modes: Equ. (6) will give a unique solution for a given transverse admittance distribution along the beam axis. For the stem configurations previously studied, the transverse admittance is constant and constant transverse admittance result in cos(nπz/l) like field distributions. Field distributions from admittance distributions inhabiting a slight modulation from a constant value will result in field distributions that are moderately modified but still similar to cosine like field distributions.
• stop-band and pass-bands: lowering the transverse admittances rises the lower pass-band. At some point the stop-band disappears and the two passbands merge.
Two different methods will be introduced in subsection A and B to minimize the transverse admittances by well distributed stems. After tuning the field tilt sensitivity all along the cavity is strongly reduced. The tuning processes will be applied to an example cavity similar to the ones for the UNILAC upgrade considering increasing cell length.
A. Tuning through T E-modes Figure 6 depicts stem configurations with quadrupolar symmetry. The capacitance C between tank wall and drift tube, together with the inductance L s of the stems, comprise a resonating structure in the T E-mode. Its frequency ω e0 can be obtained from simulating a short cavity section of few cells and periodic boundary conditions. Adjusting the stem configuration of this section to ω e0 ≃ ω 0 is equivalent to tune Y tu (ω 0 ) and Y tu (ω 0 ) to zero along the complete cavity. The alternating Vconfiguration shown in Fig. 6a features negative transverse admittances. This allows for alternating groups of stems instead of single stems, thus adjusting locally the transverse admittances to zero. Figure 12 shows stems being grouped and alternated accordingly. The stem number per group m can be determined from simulations by requiring ω e0 ≈ ω 0 . To account for increasing cell length, the stem configuration needs adoption as βλ increases. Figure 13 shows the frequency of the T E 010mode as a function of m for different βλ. m is to be chosen such that the frequency of the T E 010 -mode becomes equal to the nominal operating frequency, being 108.4 MHz in the actual example. The procedure to obtain the optimum stem configuration is:  2): βλ is fixed for the complete cavity and the frequency of the basic T E-mode is determined from simulations. If it is lower (higher) than the operational mode, the number of cells is reduced (increased) and step 1) is repeated until the frequency of the basic T E-mode comes as close as possible to the operating frequency ω 0 .
3): After finishing the design of the first cavity section, the cell length of the subsequent section is fixed and the procedure of step 2) is applied again.

4):
After finishing the last section, simulation of the complete cavity with conducting wall boundary is performed. Frequencies of modes close to the operational mode are checked as well as the field distribution of the T M 011 -mode.
The method takes advantage of short cavity simulations with high accuracy instead of many simulations of the complete cavity with reduced accuracy. Figure 14 depicts the final stem configuration, referred to as E in the following. The frequency of the T M 011 -mode is shifted from 108.4+0.5 MHz to 108.4+4.1 MHz.
FIG. 14. Stem configuration E determined through tuning the frequency of the basic T E-mode. The increase of the cell length leads to a decrease of the stem density and m decreases accordingly.

B. Tuning through T M011-mode
As mentioned in the previous section, for the constant V-configuration rotation of any pair of stems by 180 degrees will locally change the sign of the slope of the field distribution of the T M 011 -mode as shown in Fig. 15. It is from the local change of sign of the expressions for Y tu and Y tl in Equ. (8). As the frequency of the T M 011 -mode is very close to ω 0 , it can be rewritten as ω 1 = ω 0 + δω. If at some location z along the axis Y (ω 0 + δω) = 0, the field tilt of the T M 011 -mode becomes equal to zero. As Y (ω 0 ) ≃ 0 is a reasonable approximation, the field tilt of the T M 010 -mode on that location will be cancelled as well. For the ideal model the field distribution of the T M 011 -mode is E ∝ cos(πz/l). The proposed tuning method shall modify this cosine function to a function being closer to a step function, preserving the initial and final value of the original cosine function. The smooth cosine like inversion of the initial field strength at the cavity entrance to its negative value at the cavity exit will be re-shaped towards a more abrupt change of polarity close to the center of the cavity. Re-shaping is done by profiting from the fact that a 180 degree rotation of the stems changes the sign of the local slope of the T M 011 -field distribution. Proper choice of the local density of rotated stems will thus locally flatten the field distribution. The procedure to obtain the optimum stem configuration is: Simulations of a complete cavity with constant Vconfiguration are performed to obtain the field distribution of the T M 011 -mode.
2): A group of stems located not far from each end wall is rotated by 180 degrees towards the bottom of the tank. With simulations it is checked whether the bottom T M 011 field distribution is locally flattened. If not, the number of stems forming the group and/or its position is varied. This step is done from both sides of the cavity simultaneously.

3):
Step 2) is repeated with the next two groups towards the centre of the cavity.

4):
Step 3) is repeated until two flattened sections of the field connected by a step close to the cavity centre are produced as shown in Fig. 15. Figure 15 shows the T M 011 -field distribution and how it is changed by modifying the constant V-configuration of the stems. The final stem configuration, referred to as M in the following, obtained by applying this method is depicted in Fig. 16. For this stepwise flattened field distribution the frequency of the T M 011 -mode is shifted from 108.4+0.5 MHz to 108.4+4.5 MHz. The effect of the proposed tuning procedures on the operational mode is evaluated in the following section.

C. Verification through Simulations
To evaluate the efficiency of the two tuning methods, a perturbation is applied to cavities featuring designs E and M, respectively. The length of the first gap is reduced by 2 mm to 25.5 mm, causing a frequency shift ∆f of -140 kHz. Then the length of the last gap is increased by 4 mm to 53 mm such that the total frequency shift is zero. The TS of cell i is defined as where E 0i is the average field strength of cell i. Figure 17 plots the TS for different stem configurations. Both stem configurations E and M reduce significantly, i.e. they practically remove, the TS compared to the original constant V-configuration. Simulations showed that even for the extreme case that the first drift tube is removed from the cavity, the field distortion is kept below ±10%. A constant V-configuration without stabilization shows a field distortion of more than ±65% in that case. The two field-stabilizing stem configurations obtained from two tuning strategies result into a flat field distribution being very resistant to perturbations of the cavities geometry. The optimized stem configurations modify the dispersion curves of the T E-modes by increasing their slopes as summarized in Fig. 18. The nearest mode is shifted away from the operational mode.

V. CONCLUSION
Tuning of Alvarez-type cavities w.r.t. flat longitudinal field distribution with very low field tilt sensitivity was achieved in simulations. The method does not use any post-couplers which impose additional cost and rfpower losses. The required optimizations of the cavity rf-parameters were accomplished by adjusting the azimuthal orientations of the stems that bare the drift tubes. Two methods were developed: division of the cavity into short subsections and adjusting their local T E 010mode frequencies to the operational frequency. Usage of short sections significantly reduces the time for simulations. The second method transforms the longitudinal field dependency along the cavity of the T M 011 -mode from a cosine like half wave to a step like function. Both methods effectively minimize the transverse admittances and maximize the slope of the T M -modes dispersion curve, which is understood from analysis of the cavity equivalent circuit including stems and drift tubes. Both schemes lead to stem configurations that provide very low field sensitivity. The validity of the methods was benchmarked through extensive simulations.