Tunable Supermode Dielectric Resonators for Axion Dark-Matter Haloscopes

We present frequency-tuning mechanisms for dielectric resonators, which undergo “supermode” interactions as they tune. The tunable schemes are based on dielectric materials strategically placed inside traditional cylindrical resonant cavities, necessarily operating in transverse-magnetic modes for use in axion haloscopes. The first technique is based on multiple dielectric disks with radii smaller than that of the cavity. The second scheme relies on hollow dielectric cylinders similar to a Bragg resonator, but with a different location and dimension. Specifically, we engineer a significant increase in form factor for the ${\mathrm{TM}}_{030}$ mode utilizing a variation of a distributed Bragg reflector resonator. Additionally, we demonstrate an application of traditional distributed Bragg reflectors in TM modes which may be applied to a haloscope. Theoretical and experimental results are presented showing an increase in $Q$ factor and tunability due to the supermode effect. The ${\mathrm{TM}}_{030}$ ring-resonator mode offers a between 1 and 2-order-of-magnitude improvement in axion sensitivity over current conventional cavity systems and will be employed in the forthcoming ORGAN experiment.

Figure 1

$B_{\varphi }$ field distribution for an antisymmetric and symmetric supermode pair. The plots on the left show ${\mathit{B}}_{\varphi }$ as a function of $z$ for fixed radius, while the axes show negative and positive field values (in T). Both representations are computed in comsol. As the gap is increased, the lower mode (${\mathrm{TM}}_{012}$-like) can be identified as having in-phase lobes in the two dielectric pieces (symmetric supermode), while the upper mode (${\mathrm{TM}}_{013}$-like) has similar lobes, which are out of phase (antisymmetric). In the limit where the gap spacing is large, the frequency of the symmetric mode approaches the frequency of the antisymmetric mode, depending on the mode confinement within the dielectric. The symmetric supermode is sensitive to axions, as the field is largely in phase across the volume, while the antisymmetric mode is not. The sapphire disks are represented by heavy dark lines.

Figure 2

Calculated frequencies (in GHz) for the symmetric and antisymmetric supermodes (left panel, blue and orange lines, respectively), and the $C^2{V}^{2}G$ product for the symmetric supermode (right panel) vs gap size (in mm) for the disk resonator discussed in the text.

Figure 3

Density plot of transmission measurements in the proof-of-concept experiment. Lighter colors represent more transmission, whereas darker colors represent less transmission. The orange overlaid data represents the frequency of the most sensitive mode vs gap size computed via finite-element analysis. We attribute the frequency discrepancy at small gap sizes to a misalignment between the two disks, which has less impact on the modes as the gap size increases. We observe multiple mode crossing with dissimilar modes, while the higher-frequency antisymmetric supermode is too high in frequency (roughly 8 GHz, per Fig. 2) to be observed in this plot.

Figure 4

Comparison of supermode frequency tuning (left panel) and sensitivity (right panel) as a function of $x\times n/2$. The inset in the left panel shows a schematic of a multidisk resonator of $n=4$, with each disk separated by $x\mathrm{mm}$.

Figure 5

A 2D axisymmetric visualization of the $z$ component of the electric field for a ${\mathrm{TM}}_{020}$-like mode confined inside a sapphire ring, such that the Bragg effect is achieved (left panel), and a similar visualization of a ${\mathrm{TM}}_{030}$-like mode designed such that the out-of-phase electric field is contained in the sapphire ring and suppressed. We name this phenomenon the dielectric boosted axion sensitivity (DBAS) effect (right panel). The sapphire rings are represented by heavy dark lines.

Figure 6

Resonant frequencies (left panel) and geometry factors (right panel) as a function of the size of the gap between the sapphire ring and the cavity wall, computed via finite-element analysis for the first four TM modes. ${\mathrm{TM}}_{010}$ is shown in blue, ${\mathrm{TM}}_{020}$ in orange, ${\mathrm{TM}}_{030}$ in green, and ${\mathrm{TM}}_{040}$ in red. The sapphire ring is of fixed dimensions (outer radius of 24.41 mm, thickness of 3.24 mm, and height of 31.98 mm). The TM modes become Bragg confined modes in the regions where the geometric factor is enhanced and the field is antiresonant in the reflectors. These regions can be seen to be quite broad over gap-size variations. Experimental measurements of mode frequencies from cryogenic proof-of-concept measurements are overlaid on the left plot.

Figure 7

Representation of the $E_z$ component of the ${\mathrm{TM}}_{030}$ mode as a function of radial distance from the center of the cavity. The mode takes the form of a Bessel $J$ function.

Figure 8

$E_z$ field distribution for the two modes discussed in the text as the gap between the sapphire rings increases. Redder colors represent higher positive $E_z$ values, whereas bluer colors represent higher negative field values. The lower mode is a ${\mathrm{TM}}_{030}$-like mode, with high axion sensitivity, whereas the higher mode is a ${\mathrm{TM}}_{031}$-like mode with no axion sensitivity.

Figure 9

(Left panel) Frequency (in GHz), and (right panel) $C^2{V}^{2}G$ product vs gap size (in m) for the ideal DBAS resonator discussed in the text.

Figure 10

$C^2{V}^{2}G$ product versus thickness of the dielectric region as a fraction of cavity radius, with each point scaled to 5 GHz for comparison. The gap size and other parameters are kept constant in this sweep. As shown, the optimum lies near 0.15, which is close to what is predicted in the text for the DBAS effect.

Figure 11

(Left panel) Frequency (in GHz), and (right panel) $C^2{V}^{2}G$ product vs rod distance from the center of the cavity (in m) for the conducting rod-tuned resonator discussed in the text.