A tunable resonator enabled by a soft impedance surface

We present a design for a cylindrical cavity resonator with a resonant frequency ( TM010 ${\text{TM}}_{010}$ mode) that can be tuned by increasing the separation between the cavity's main body and a discrete lid. This is made possible by incorporating a soft impedance surface into the lid. Tuning between 5 and 6 GHz is achieved by increasing the lid‐to‐cavity separation from 0 to 1.5 mm. We will also discuss the field distribution within the cavity, and use it to explain how the cavity works, and limitations on the performance.


| INTRODUCTION
Cavity resonators are radio frequency (RF) components that are used in a wide range of applications, including particle accelerators, 1 dark matter detectors, 2 transducers, 3 and for electron paramagnetic resonance measurements. 4Usually they are either cylindrical cavities tuned by moving rods to alter the internal field distribution, or smaller reentrant cavities, 5,6 which are tuned by varying the capacitance between a central post and the lid.
In this paper we will present a design for a cylindrical cavity, where the resonant frequency of the TM 010 mode can be tuned over a 20% bandwidth by adjusting the separation between the main body of the cavity and a discrete lid.RF energy is contained within the cavity by utilizing a soft impedance surface.We developed this cavity as a prototype cavity for a dark matter (axion) detection experiment; where the TM 010 can resonantly enhance the photons produced when axions convert into photons in the presence of a magnetic field via the inverse Primakoff Effect. 7,8The soft impedance surface could also be useful for rotary waveguide joints, and the fabrication of corrugated feedhorns. 9n Section 2 we will discuss how the soft impedance surface has been applied to the cavity by first examining its use on a section of circular waveguide; showing how its performance is superior to that of a conventional waveguide choke.In Section 3 we discuss the design of the cavity, and use simulations of the electromagnetic field to explain the dependence of the tunable bandwidth on several design parameters.In Section 4 we discuss the manufacturing methods, and present resonant frequency and Q-factor measurements for aluminum and copper cavities.

| SOFT IMPEDANCE SURFACE
The concept of a soft impedance surface, also known as an electronic or photonic bandgap has been studied for several decades. 10,11They exploit periodic structures to create regions that stop the transmission of electromagnetic waves along a surface, and are typically formed using a repeating pattern of pins.Owing to their usefulness in allowing current lines to be cut, 12 we initially explored using an array of pins to create a re-entrant cavity with a discrete lid.However, simulations showed that bringing the pins closer together improved the performance of the cavity, with circular ridges offering optimum tunability.
Using Ansys' High-Frequency Structure Simulator, the stopband of a conventional electronic bandgap structure is calculated using a square unit cell and the eigenmode solver, with the propagation constant (β) of the supported modes given by where ϕ Δ is the phase difference between the master and slave boundaries, d is the width of the pin and ds is the distance between pins.However, for circular ridges, since HFSS does not support master and slave boundaries on walls with different areas, we approximate the radial symmetry by using a rectangular cell with the pin replaced by a ridge.The master and slave boundaries are applied to the sidewalls that face the radial dimension, whilst perfect magnetic conduction boundaries are applied to the walls that face along the azimuthal direction.Perfect electrical conductor boundaries remain on the top and bottom walls.The unit cell can be seen in Figure 1A.To avoid exciting additional modes the ridge length (w) should not be too large, with ∕ w d ds = ( + ) 2 found to be acceptable.The resulting stopband can be seen in Figure 1C.
The effectiveness of the bandgap can be explored by simulating a length of circular waveguide that is assumed to be made from a stack of disks.One of the disks can be seen in Figure 1B, and the results of the simulation can be seen in Figure 1C. Figure 1C shows that the bandgap is effective at preventing RF leakage to almost the predicted 8.2 GHz, and we believe that the onset of leakage at 7.9 GHz is due to the effect of fringing fields that are not considered in the stopband simulation.Also shown in Figure 1C is the simulated performance of a conventional waveguide choke (designed for 6 GHz), showing that it is not effective at this separation.

| DESIGN AND SIMULATION
A sketch of the cross section of the cavity, with the main dimensions can be seen in Figure 2A.R c the cavity's radius sets the minimum resonant frequency, which is given by where c is the speed of light.R 1 is the radius of the lid's central boss, which is slightly larger than R c to ensure that the cavity is closed for zero air gap.g the air gap tunes the cavity.Simulations in HFSS showed that the degree of tunability was influenced by the cavity's height (H c ), and the width of the initial groove (s), with taller cavities possessing less tunability, and narrower grooves resulting in increased tunability.This behavior can be seen in Figure 2, and can be explained by considering the field distribution inside the cavity, which can be seen in Figure 3. Towards the center of the cavity the standard TM 010 mode that is found in cylindrical cavities is present.However, at the top, and towards the edge there is a null in the electrical field that also corresponds to a reversal in the direction of the field.As the air gap is increased, this reversal point moves towards the center of the cavity, squeezing the TM 010 mode into the volume of a truncated cone and reducing the effective radius.Consequently, for taller cavities, the truncated becomes less significant, and the effective radius is closer to that of a cylindrical cavity.Similar behavior was observed for the width of the initial groove, with simulations showing that for an air gap of 1 mm and s= 2 mm, the position of the null for a 15-mm tall cavity was approximately 13.5 mm from the center of the cavity, whilst for s = 7 mm it was approximately 17.5 mm from the center.Similar simulations were carried out for a standard waveguide choke ring and no tunability was observed.

| MEASUREMENTS
Aluminum and copper prototypes were fabricated via computer numerical control machining.The aluminum cavity was machined from a single piece, whilst the copper cavity was machined as two pieces that were silver soldered together.Two designs of were also explored, with the copper lid incorporating an extra lip that shorts the cavity at the outer edge.RF was coupled into the cavity using a pair of probes that were fabricated from female SMA connectors, with a short length of copper wire inserted into the hole for the central conductor.Images of the cavities can be seen in Figure 4.
The lids were moved mechanically using a Thorlabs MS1S Single Axis Translation Stage, and the micrometer was used to measure the size of the air gap.
The resonant frequencies of the cavities were measured using a Keysight PNA-X, and the results for the aluminum cavity can be seen in Figure 5. Figure 5 shows that the achieved tuning range is fractionally smaller than that predicted from simulations.Similar performance was also observed for the all copper cavity, although in this case the minimum frequency was higher at approximately 5.2 GHz, which we believe is due to the cavity not fully closing; due to warping of the thin plate that occurred during the soldering process.Indeed judging zero air gap is difficult, and this may well account for the offset between the simulated and measured results.
The aluminum cavity, with a coupling that was close to critical has a Q-factor of approximately 1600, whilst the copper cavity has a Q-factor of approximately 2200.

| CONCLUSION
In this paper we have presented a cylindrical resonant cavity that is tunable from approximately 5-6 GHz by opening up a small gap between the main body of the cavity and the lid.RF leakage is prevented by using a series of rings that create a soft impedance surface.The soft impedance surface also provides the tuning mechanism by interacting with the field within the main cavity, and confining the TM 010 to the volume of a truncated cone with a varying upper radius.

F
I G U R E 1 (A) The unit cell, shaded walls are the master and slave boundaries.(B) One of the platelets used in the waveguide simulation.(C) The insertion loss of a length of waveguide made from 10 sections, 1.5 mm apart, with either a conventional choke (2 mm wide with ∕ λ 4 = 16 g mm) or a soft impedance boundary.The simulated stopband is also shown.

F
I G U R E 2 (A) Sketch showing the dimensions of the cavity.The final dimensions were as follows, R c = 23.5 mm, R 1 = 24 mm, H c = 15 mm, s = 3.6 mm, h = 7 mm, and d = ds = 12 mm. (B) The effect of changing the height of the cavity, and (C) the width of the inner groove on the resonant frequency.F I G U R E 3 Field distribution within the cavity for different air gaps (0.5, 1, and 1.5 mm).As the air gap opens up the null and field reversal point at the top of the cavity (denoted by *) moves closer to the center of the cavity.Arrows indicate the direction of the electric field.Field strengths are nominal.F I G U R E 4 (A) The aluminum resonating cavity with input and output probes, (B) the aluminum cavity lid, nylon alignment pins are used to align the cavity and the lid, (C) the copper cavity lid with extra outer lip, and (D) the mounted resonator.F I G U R E 5 The tunability aluminum cavity, dashed lines show the simulated performance.