Creating Exact Multipolar Fields with Azimuthally Modulated RF Cavities

The Panofsky-Wenzel theorem therefore states that only the longitudinal electric field of the mode need be determined in order to calculate the effect of the transverse forces that act on a particle beam passing through a cavity. The general form of the electric field, $\vec{E}(r,\theta,z)$, of a mode supported by an RF cavity of any design is derived in Ref. [24]. This general form can be simplified to determine the specific form of the longitudinal electric field in azimuthally modulated cavities by applying the boundary conditions at the endpoints and cavity surface. Applying the endpoint boundary condition, we find that modes of distinct longitudinal order $p$ must independently satisfy the boundary conditions. Applying the surface boundary condition, we find that the allowed modes belong to one of two familiar sets: either TM modes with no longitudinal magnetic field or TE modes with no longitudinal electric field. In this paper, we only study TM modes with $p=0$: TE modes are ignored as they are less frequently used in particle accelerators and their effect on beam dynamics cannot be analysed using the Panofsky-Wenzel theorem.

Although numerical methods must generally be used to solve Eq. II.3 for all $\theta$, insight into the properties of the azimuthally modulated RF cavities that support TM_{M}η0 modes can be gained through analysis using mathematical and graphical methods.

There are a number of important, known results for the TM${}_{m_{1}n0}$ modes [12]. Firstly, because all Bessel functions have $(n-1)$ turning points in the range [0, $j_{m_{1}n}$] and the field varies with the azimuth simply as $\cos{(m_{1}\theta-\phi_{m_{1}})}$, the integer $n$ is the number of poles (that is minima or maxima) of $E_{z}$ along any radial line that connects the centre of the cavity to its surface, except the nodal lines orientated at $\theta_{q}=(2q+1)\pi/2m_{1}+\phi_{m_{1}}$ ($q\in\mathbb{Z}$). Secondly, the $m_{1}\neq 0$ modes are degenerate because the boundary condition in Eq. II.4 is satisfied for all values of $\phi_{m_{1}}$. In contrast, the TM_0n0 modes are not degenerate. Thirdly, as $j_{m_{1}n}$ is unique for all distinct values of $m_{1}$ and $n$, the resonant frequencies of each of the TM${}_{m_{1}n0}$ modes of distinct $m_{1}$ and $n$ are unique. Thus to use the TM_mn0 modes to introduce numerous multipoles at the same frequency, multiple pillbox cavities of different radii would have to be used. Finally, the TM₀₁₀ mode is the fundamental TM_mnp mode with the lowest resonant frequency.

We can also use a graphical method to argue that TM${}_{m_{1}n0}$ modes will be supported by cavities with a circular cross-section. Considering the 3 GHz TM_1n0 modes with $\tilde{g}_{1}=1$ and $\phi_{1}=0$, Eq. II.4 states that the $r_{0}^{(n)}(\theta_{0})$ solutions at a particular angle $\theta_{0}$ occur where, if plotted as a function of $r$, $J_{1}(kr)\cos{\theta_{0}}$ intersects the horizontal $r$-axis. Figure 1 plots $J_{1}(kr)$ (the case where $\theta_{0}$ = 0) for $r=[0,25]$ cm. The first three positive intersections with the $r$-axis are indicated by the dashed black, blue, and cyan lines that correspond respectively to the $r^{(n)}(0)$ solutions with $n$ = 1, 2, and 3.

We could next consider plotting $J_{1}(kr)\cos{\theta}$ for many $\theta$ values and determining where the intersections lie for each $\theta$. This is not necessary, however, because the cosine term in $J_{1}(kr)\cos{\theta}$ acts as a $\theta$-dependent amplifying term on the Bessel function and thus the locations of the intersections with the $r$-axis will not change. As a result, we conclude that the $r_{0}^{(n)}(\theta)$ solutions are constant with respect to changes in $\theta$ and so plotting any $r_{0}^{(n)}(\theta)$ solution in Cartesian space plots a circle. This graphical argument can be generalised to argue that TM${}_{m_{1}n0}$ modes of any $\tilde{g}_{m_{1}}$ and $\phi_{m_{1}}$ will be supported by a circular pillbox.

We now investigate the properties of the TM${}_{\{0,m_{1}\}\eta 0}$ modes and the corresponding azimuthally modulated cross-sections that support them. We begin with general mathematical analysis.

From Eq. II.2, the subset of TM${}_{\{0,m_{1}\}\eta 0}$ modes with $\phi_{m_{1}}=0$ have a longitudinal electric field of the form:

and from Eq. II.3, must satisfy the boundary condition:

The periodicity of the $\cos{m_{1}\theta}$ function in Eq. II.7 means that both the longitudinal electric field of a TM${}_{\{0,m_{1}\}\eta 0}$ mode and the corresponding azimuthally modulated cross-section have $m_{1}$-fold rotational symmetry. Additionally, $r_{0}^{(\eta)}(0)$ and $r_{0}^{(\eta)}(\pi/m_{1})$ are turning points of opposite forms in the cross-section and so if $r_{0}^{(\eta)}(0)$ is a maximum then $r_{0}^{(\eta)}(\pi/m_{1})$ is a minimum, or vice versa.

As per the TM_0n0 modes in the circular pillbox cavity, the TM${}_{\{0,m_{1}\}\eta 0}$ modes supported by azimuthally modulated cavities are also non-degenerate with no same-order mode. Furthermore, by the Courant nodal domain theorem [25], the TM${}_{\{0,m_{1}\}10}$ modes are the fundamental TM modes in the azimuthally modulated cavities that support them.

Additionally, we note that if the ratio of the transverse multipolar component to the monopolar component is zero, $\tilde{g}_{m_{1}}/\tilde{g}_{0}=0$, the form of $E_{z}$ in Eq. II.6 is equivalent to that of a TM_0n0 mode. Such modes are supported by a cavity with a circular cross-section, as discussed in Sec. II.2. The cross-section of the cavity that supports a TM${}_{\{0,m_{1}\}\eta 0}$ mode for a small ratio of transverse multipole to monopole, $\tilde{g}_{m_{1}}/\tilde{g}_{0}<<1$, will therefore be perturbed only slightly from circular. Taking the other extreme whereby the transverse multipole to monopole ratio tends to infinity, $\tilde{g}_{m_{1}}/\tilde{g}_{0}\rightarrow\infty$, the cavity cross-sections do not tend back to circular but instead tend to unique, non-circular shapes. This asymmetry arises because the TM${}_{m_{1}n0}$ modes are degenerate whereas the TM${}_{\{0,m_{1}\}\eta 0}$ modes are non-degenerate for all ratios of $\tilde{g}_{m_{1}}/\tilde{g}_{0}$.

Finally, we consider the azimuthally modulated cross-sections that support general TM${}_{\{0,m_{1}\}\eta 0}$ modes for any value of $\phi_{m_{1}}$. Introducing a $\phi_{m_{1}}\neq 0$ term into Eq. II.6 and Eq. II.7, we find the set of general cavity cross-sections are identical to the subset of cavity cross-sections with $\phi_{m_{1}}=0$, except for a physical rotation about the $z$-axis of $\phi_{m_{1}}/m_{1}$.

Having determined the properties of azimuthally modulated RF cavities that support the TM${}_{\{0,m_{1}\}\eta 0}$ modes in the previous subsection, we now examine the azimuthally modulated RF cavities that support the TM${}_{\{m_{1},m_{2}\}\eta 0}$ modes composed of two transverse multipoles.

Using Eq. II.2, the TM${}_{\{m_{1},m_{2}\}\eta 0}$ modes composed of two different, transverse multipoles with $m_{2}>m_{1}>0$ have a longitudinal electric field of the form:

Without loss of generality, the corresponding set of azimuthally modulated cavity shapes can be determined from Eq. II.3 with $\phi_{m_{1}}=0$:

The cross-sections that solve Eq. II.12 and support TM${}_{\{m_{1},m_{2}\}\eta 0}$ modes require further consideration compared to those that support TM${}_{\{0,m_{1}\}\eta 0}$ modes for four interlinked reasons:

• •

  Solutions do not necessarily have an $m_{1}$-fold symmetry but instead a rotational symmetry that varies depending upon the values of $m_{1}$, $m_{2}$ and $\phi_{m_{2}}$. As a result, $\eta$ has a more specific definition for TM${}_{\{m_{1},m_{2}\}\eta 0}$ modes.

• •

  A practical RF cavity must have $r_{0}^{(\eta)}(\theta)>0$, however, $r_{0}^{(\eta)}(\theta)=0$ is a valid solution to Eq. II.12 for all $m_{1}$ and $m_{2}$. We therefore define a TM${}_{\{m_{1},m_{2}\}\eta 0}$ mode as forbidden if $r_{0}^{(\eta)}(\theta)=0$ at any angle $\theta$.

• •

  Solutions are not necessarily bounded within the interval $r=[j_{m_{2}(x-1)}$, $j_{m_{2}x}]/k$ due to both terms in Eq. II.12 having differing orders of cosine dependence.

• •

  A practical RF cavity must be closed with $r^{(\eta)}(0)=r^{(\eta)}(2\pi)$, however, solutions of particular $m_{1}$, $m_{2}$ and $\phi_{m_{2}}$ do not satisfy this condition. We define such TM${}_{\{m_{1},m_{2}\}\eta 0}$ modes as spiral.

Appendix A derives why certain modes are forbidden or spiral. Section II.4.1 presents an example of a forbidden mode and Sec. II.4.2 an example of a spiral mode.

TM${}_{\{m_{1},m_{2}\}\eta 0}$ modes can also be conditional whereby the cross-section is discontinuous and a hybrid solution needs be considered if the ratio of the transverse multipolar coefficient of order $m_{1}$ to the transverse multipolar coefficient of order $m_{2}$ (or vice versa) exceeds a critical value. A TM${}_{\{m_{1},m_{2}\}\eta 0}$ mode will be conditional if, for any $\theta=\theta_{0}$, there is more than one root to the $m_{1}^{\textrm{th}}$ order Bessel function within the $[j_{m_{2}(x-1)},j_{m_{2}x}]/k$ interval that the $r_{0}^{(\eta)}(\theta_{0})$ solution exists in.

Equation II.3 can also be numerically solved to determine the azimuthally modulated cross-sections that support TM_{M}η0 modes with three or more multipolar components. General mathematical analysis of the properties of these modes and the corresponding cross-sections that support them is challenging. The concepts of certain modes being conditional, forbidden or spiral, however, generalises to TM_{M}η0 modes with any number multipolar content and provides rationale for understanding the numerical solutions to Eq. II.3 as well as helping to inform the scope and limitation of a general azimuthally modulated RF cavity design.

Figure 9 shows the E-field distribution of a critically-coupled, single-cell, azimuthally modulated RF cavity system designed to support a 3 GHz TM_{0,2}20 mode with a quadrupole to monopole ratio of 5.041. Instead of using a single-port coupling system which would introduce a dipolar component into the field of the RF cavity, a dual-port coupling system was used in conjunction with a T-junction waveguide design to split the input power equally into the two waveguide arms. The dual-port coupler introduces an additional quadrupolar component into the mode. Helmholtz decompositions of simulations showed that by solving Eq. II.3 for the enclosed RF cavity shape that supports a 3 GHz TM_{0,2}20 mode with $\tilde{g}_{2}/\tilde{g}_{0}$ = 4.032 and $\phi_{2}=0$, and using this shape to create the prototype in Fig. 9 which includes the dual port coupler and holes at the positions of the poles, the resultant field had a quadrupole to monopole ratio of $\tilde{g}_{2}/\tilde{g}_{0}$ = 5.041 and negligible contributions from higher order multipoles. The $\eta=2$ mode was chosen over the $\eta=1$ mode in order to give more measurement points to consider and compare.

The RF cavity was designed as two identical halves to be sealed together and was milled out of aluminium. Seven 10 mm diameter holes were drilled into the RF cavity to create holes with centres at the locations of the seven poles in the TM_{0,2}20 mode. One half the RF cavity system is photographed in Fig. 10.

Measurements of the prototype were taken using a Vector Network Analyser (VNA) connected to a matched waveguide adaptor secured to the top of the cavity waveguide port via coaxial cable. The apparatus was calibrated at the end of the coax, which excluded the matched waveguide adaptor from calibration. To ensure this does not impact results, two cases are compared in Fig. 11: the frequency response of shorting the calibrated coax, and of attaching the waveguide adaptor to the calibrated coax and clamping an aluminium plate to it. The maximum magnitude in the difference of the impedance measurements between the two cases is 0.2 $\Omega$. This is of the order of 100 times smaller than the changes in impedance measured in the bead pull testing and that used for calculating Q-factors. In addition to this, measurements do not give any evidence for resonances in the waveguide adaptor and the external Q-factor does not change significantly between simulations without the waveguide adaptor and measurements with the waveguide adaptor. We thus conclude that the waveguide adaptor’s effect on measurements can be neglected.

The resonant frequency of the RF cavity system, $f_{0}$, and its intrinsic, $Q_{0}$, external, $Q_{e}$ and loaded, $Q_{L}$, Q-factors can be calculated from measurement of the variation in the S-parameter $S_{11}$ with driving frequency and using:

where $f$ is the driving frequency and $\beta$ is the coupling coefficient. It should be noted that at resonance $S_{11}$ is purely real.

Figure 12 is a Smith chart that shows the measured variation in $S_{11}$ for driving frequencies in the range $f_{0}\pm 1$ MHz for five different simulation and experimental measurements. The Smith chart maps the impedance, $Z$, onto the $S_{11}$ plane and provides a method for visualising $\beta$ [26].

The blue line of Fig. 12 represents the simulation output of $S_{11}$ of the RF cavity system modelled as perfectly smooth copper; the electric field from this simulation was shown in Fig. 9. From this data, we calculate $f_{0}$ = 2.99914 GHz, $Q_{0}$ = 20830, $Q_{e}$ = 20050 and so the system is critically coupled as $\beta\simeq 1$. The RF cavity coupling ports were optimised for copper, while the prototype was milled out of aluminium to reduce cost. The red line in Fig. 12 shows the simulation output for perfectly smooth aluminium and the impact of the choice of material on $\beta$: we calculate $f_{0}$ = 2.99917 GHz, $Q_{0}$ = 16880 and $\beta=0.831$. The yellow line represents the measurement of the prototype, and we calculate that $f_{0}=2.99914$ GHz, $Q_{0}=9120$ and $\beta=0.410$. The prototype is thus significantly undercoupled compared to the perfectly smooth simulation.

This discrepancy in $\beta$ is likely due to two effects. Firstly, the surface roughness due to machining of the aluminium which leads to increased resistive losses [27]. The RMS surface roughness was measured with a white light interferometer to be 0.33 $\mu$m. The purple line shows the simulation output when this level of roughness is included - $f_{0}$ = 2.99955 GHz, $Q_{0}$ = 15150 and $\beta=0.750$. The second effect is RF leakage due to an imperfect seal between the two halves of the prototype. This can be modelled by uniformly separating the two cavity halves: the green dashed line shows that a 0.33 $\mu$m surface roughness and 3.5 $\mu$m uniform air gap between the two halves replicates the coupling coefficient of experimental measurements - $f_{0}$ = 2.99955 GHz, $Q_{0}$ = 8490 and $\beta=0.410$.

In reality the separation between the two halves of the cavity will not be uniform, however, this would be difficult to model. Simulations show the prototype is particularly sensitive to gaps near the coupling between the waveguide and the cavity and a separation on the order of microns is feasible given the distribution of the bolts used for tightening and the presence of deeper scratches on the aluminium surfaces that are bolted together.

Despite the RF cavity system being significantly undercoupled, the electric field of the RF cavity can be determined using the bead pull technique [28, 29, 30]. This method uses the result from cavity perturbation theory [31] that when a dielectric bead, small enough such that the EM fields are approximately constant over it, is introduced into an RF cavity at a position $(r,\theta,z)$:

where $\Delta f$ is the change in resonant frequency caused by introducing the perturbing object and $E_{0}(r,\theta,z)$ is the magnitude of the electric field at that position. In our experiments, we used a spherical, plastic bead with a 2.2 mm diameter as the perturbing object.

The VNA cannot be used to measure directly the change in resonant frequency of the RF cavity caused by the perturbing object at a high enough sample rate. An indirect measurement is therefore required. Two possible parameters to use for this indirect measurement are the change in phase of $S_{11}$ or $|\Delta Z|$, where $|\Delta Z|=|Z-Z_{\textrm{res}}^{\textrm{unpert}}|$ and $Z^{\textrm{unpert}}_{\textrm{res}}$ is the impedance of the unperturbed cavity at the resonant frequency. Figure 13 shows experimental measurements of the variation in $\tan{(\arg{(S_{11})})}$ (top) and $|\Delta Z|$ (bottom) of the unperturbed cavity with $f$. The dashed lines are linear fits between the respective parameter and $f$, fitted to and extrapolated from small $\Delta f$. The 2.2 mm diameter perturbing bead caused a maximum change in resonant frequency of $|\Delta f|$ = 0.079 MHz and the inset of Fig. 13 zooms in on this region. It shows $|\Delta Z|$ varies linearly with $f$ to within 0.3 %, compared to $\tan{(\arg{(S_{11})})}$ which diverges by up to 20 %.

We can therefore determine the electric field at a position $(r,\theta,z)$ in the RF cavity by measuring, at a fixed driving frequency, the change in impedance caused by the bead at that position and using the relationship:

To carry out a bead pull measurement, we threaded kevlar wire through one of the holes before centering it using a digital motor system. The 2.2 mm diameter, dielectric bead knotted onto the wire was then pulled through the cavity at a speed of 3 mm/s using a longitudinal motor. The VNA was set up to continuously measure the impedance of the RF cavity at the driving frequency of 2.99914 GHz and sample rate of over 50 points per second. The change in impedance of the RF cavity was measured for the bead being pulled through all 7 holes in both longitudinal directions.

The data was exported and analysed in MATLAB [32]. Data collected when the bead was outside the beam pipes, that is $|z|>38.1$ mm where $z$ is the longitudinal position relative to the centre of the RF cavity, was removed from the data set. A line of best fit between the first and last 50 data points was then used to remove the small linear drift in impedance over time we observed.

Figure 14 compares the longitudinal electric field measurement along each of the seven poles from a CST simulation of the cavity system to the experimental bead pull measurements, where we assume $|E_{0}|=|E_{z}|$ for the measurements. Simulation data is plotted as full lines and experimental data as dashed. The different colours correspond to measurements along the different poles: blue denotes the single central pole, red the two inner horizontal ($x$) poles, green the two outer horizontal ($x$) poles, and black the two vertical ($y$) poles. To normalise the simulation data, all data points were divided by the maximum electric field in the $y$ pole. To normalise the experimental data, all data points were divided by the ratio of the mean value of the median experimental electric field measurements between $|z|<1$ mm along each pole to the mean value of the maximum simulated electric field measurements along each pole. The similarity between experimental and simulation results indicate that the cavity is symmetric in the horizontal, vertical, and longitudinal directions, as expected.

Table 1 presents a quantitative comparison between the experimental and simulation results with the mean of the electric field measurement between $|z|<1$ mm shown alongside the standard deviation of the measurements in that range. $E_{z}$ is expected to be flat in this range with a standard deviation below three decimal places and the results show the experimental data is consistent with simulation. It should be noted that errors arising from inaccuracies aligning the wire to the centre of the holes or from divergences from the assumed linearity in Eq. IV.3 have not been included in this analysis.

The data collected thus support that the electric field of the RF cavity is as designed, within error. Therefore we have used successfully the presented theory, in conjunction with 3D simulations to determine the quadrupolar term introduced by the dual-port coupler, to design and measure the field of an azimuthally modulated RF cavity that supports a TM_{0,2}20 mode with a quadrupolar to monopolar ratio of 5.041.