# Wire metamaterial filled metallic resonators

Rustam Balafendiev<EMAIL_ADDRESS>School of Physics and
Engineering, ITMO University, 197101 St. Petersburg, Russia  Constantin
Simovski<EMAIL_ADDRESS>Department of Electronics and
Nanoengineering, Aalto University, School of Electrical Engineering, P.O. Box
13000, 0007 Aalto, Finland  Alexander J. Millar<EMAIL_ADDRESS>The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics,
Stockholm University, Albanova University Center, SE-10691 Stockholm, Sweden
Nordita, KTH Royal Institute of Technology and Stockholm University,
Roslagstullsbacken 23, 10691 Stockholm, Sweden  Pavel Belov
<EMAIL_ADDRESS>School of Physics and Engineering, ITMO University,
197101 St. Petersburg, Russia

(August 27, 2024)

###### Abstract

In this work we study electromagnetic properties of a resonator recently
suggested for the search of axions — a hypothetical candidate to explain dark
matter. A wire medium loaded resonator (called a plasma haloscope when used to
search for dark matter) consists of a box filled with a dense array of
parallel wires electrically connected to top and bottom walls. We show that
the homogenization model of a wire medium works for this resonator without
mesoscopic corrections, and that the resonator quality factor $Q$ at the
frequency of our interest drops versus the growth of the resonator volume $V$
until it is dominated by resistive losses in the wires. We find that even at
room temperature metals like copper can give quality factors in the thousands
— an order of magnitude higher than originally assumed. Our theoretical
results for both loaded and unloaded resonator quality factors were confirmed
by building an experimental prototype. We discuss ways to further improve wire
medium loaded resonators.

††preprint: APS/123-QED††preprint: NORDITA-2022-013

##  I Introduction

Despite wire media (WM) being one of the first metamaterials [1], their full
potential has yet to be explored. The unique properties of WM, an indefinite
dielectric media with strong spatial dispersion, has led to it being important
in novel applications from the radio frequencies to the optical range. Simple
wire media (uniform arrays of parallel wires) offer strong anisotropy in one
frequency range and nearly zero effective permittivity in another range,
whereas double wire media grant amazingly high values of the permittivity. In
the overview work [2] one may find a long list of WM applications known up to
2012 – from far-zone transport of near field images to drastic enhancement of
thermophotovoltaic generators. Since that time simple WM of semiconductor
nanowires have found applications in modern solar panels (so-called black
silicon [3]), microwave antennas with improved patterns [4], and as frequency
selective thermal emitters for radiative cooling of microlasers and in near-
field thermophotovoltaic systems [5, 6].

However, the use of WM in resonant systems has so far been very sparsely
studied. One can mention only works [7, 8], where finite samples of nanowire
media were suggested as radiating resonators granting a huge Purcell factor to
fluorescence emitters in the mid-IR and near-IR bands, respectively. However,
WM resonators are not only capable of enhancing molecular emission. One
exciting possibility is to use a finite sample of WM in order to implement a
novel type of microwave resonator called a plasma haloscope in the seminal
work of Ref. [9] to search for dark matter. The resonator suggested in this
theoretical paper is a metal box filled with a simple wire medium. The
governing idea of this resonator is based on the behavior of the WM as an
effective plasma.

At its plasma frequency the effective permittivity of WM crosses zero, which
enables the epsilon-near-zero (ENZ) regime greatly increasing the wavelength
of light in the medium. In contrast to the case of optical frequencies, where
a number of natural materials such as semiconductors and metals can be used
for ENZ, at radio frequencies this can only be achieved with artificial
materials, and the WM was chosen in [9] as a straightforward candidate. The
behavior of WM in the ENZ regime has been partially studied: microwave WM in
this regime offered energy tunneling through subwavelength channels [10] and
excellent directionality of antennas [11, 12, 4], whereas infrared WM granted
the unbounded spatial spectrum of eigenmodes promising the giant enhancement
of Raman radiation and fluorescence of molecules located in such media [13].
However, the idea of Ref. [9] exploited the increase of the effective
wavelength that enables at a given frequency a very low mode number for a very
big resonator, much bigger than the empty metal cavities having the
fundamental mode at the same frequency. The low-frequency resonance for a big
resonator together with the high uniformity of the eigenmode field (at much
larger scales than a traditional cavity allows) makes the plasma haloscope
promising for the search of axions, hypothetical particles of dark matter.

The composition of dark matter is one of the most enduring problems in modern
cosmology. While historically considered the second most promising candidate,
the axion has risen in prominence due to the continued non-detection of weakly
interacting massive particles. Originally proposed to solve the Strong CP
problem, the mysteriously precise conservation of Charge-Parity symmetry in
the Strong interaction [14, 15, 16], the axion can be produced non-thermally
in the early Universe to provide a natural dark matter candidate [17, 18, 19,
20, 21, 22].

Due to an anomalous coupling to electromagnetism, the axion mixes with photons
when a strong external magnetic field is applied. This mixing generates a
small electric field, oscillating at a frequency corresponding to the axion
mass and spatially constant on scales smaller than the de Broglie wavelength
(which, due to the small escape velocity of the galaxy is approximately 1000
times larger than the Compton wavelength). Traditionally, this effect has been
used to search for axions via the excitation of cavities [23]. The coupling
between the axion and the experiment is determined by the overlap of cavity
mode with the axion field, with the highest overlap occurring for the most
homogeneous mode [23]. Since the exact value of the axion mass is unknown, the
experimental setup has to be tunable over a wide range of frequencies, as well
as being large enough to provide a reasonable signal, with power scaling with
the volume and quality factor of the resonator [23].

While making a large microwave resonator is straightforward at the frequencies
on the order of hundreds of MHz to GHz [23, 24, 25], the axion may have
significantly higher masses, with some recent calculations predicting a
frequency of $16\pm 1.5$ GHz [26]. At these frequencies a conventional
resonator has dimensions of a few centimeters, leading to both a massive loss
of signal power and increased mechanical complexity. As a result, novel
experimental methods are needed, with recent proposals using multiple or
coupled cavities [27, 28, 29, 30], cavities with dielectric inserts to modify
the mode structure [31, 32, 33, 34, 35], or even abandoning a traditional
resonator in favor of a mirror [36, 37, 38, 39, 40] or of an array of large
dielectric disks [41, 42, 43, 44].

As shown in Ref. [9], the plasma haloscope allows one to overcome the mass
difference between the axion and the photon, allowing for much larger resonant
systems with close to homogeneous mode structures. While natural plasma
operable at cryogenic temperatures with low loss and a controllable plasma
frequency in the microwave regime does not exist, WM allow for a bespoke
plasma to be made. WM made of a non-magnetic metal such as copper can operate
in high DC magnetic fields and at low temperatures. They exhibit low loss,
and, importantly, can be controlled mechanically (e.g., moving the wires) to
allow simple tuning mechanisms.

However, previous work (Refs. [9, 45, 46]) were based on a simple effective
medium approach, and did not calculate the properties of such a medium from
first princples. First of all, WM are spatially dispersive, and it is not
evident how the effective permittivity known for an unbounded WM is applicable
to a finite sample. Is it mesoscopic (sensitive to the sample sizes and to the
surroundings, i.e., the walls being metal or not)? Mesocopy was not considered
in Refs. [9, 45, 46] but it is well known that mesoscopy exists for a layer of
low-loss WM located in a dielectric host or in free space. In this situation
the effective permittivity of WM is useless without the so-called additional
boundary conditions (ABCs) [47]. The problem of boundary conditions in
metamaterials can be highly nontrivial as evidenced, for instance, by the
recent rigorous analysis of boundary conditions in layered metamaterials [48].
While in Refs. [7, 8] finite samples of WM were successfully modeled without
ABCs, this was possible only because they were lossy (as was shown in work
[49], for lossy WM layers ABCs are not needed). In our case, the WM sample
must have low losses. Thus, the applicability of the homogenization model has
to be checked in view of the possible mesoscopy. Moreover, Refs. [9, 45, 46]
assumed that the volume of the system $V$ and quality factor $Q$ were
unrelated, an important assumption as the signal strength of an experiment
would be proportional to $QV$. Wires are not perfectly conducting and by
filling the cavity with them we bring significant losses into the resonator.
This effect can be taken into account by a homogenous effective medium
approach, but the size of the effect has not been estimated. Also, there may
be additional losses at the places where the wires are connected to the cavity
walls. This possible effect is not taken into account by the homogenization
model. Together, the WM sample inside the cavity must downgrade its quality
factor, but by how much? How does this downgrading counteract the gain granted
by the enlarged volume $V$?

In our work, we estimate these issues, providing a comparison of the
analytical model with full-wave simulations and measurements. Our results are
positive: we demonstrated the feasibility of a large-scale haloscope. We
analytically derived the quality factor of the WM loaded cavity and made
analytical calculations of the mode frequencies and mode fields. Comparison
with the full-wave simulations has shown the analytic model works without any
mesoscopic corrections if the metal walls are separated from the wires by one
half of the WM internal period. We found that even at room temperatures one
can have $Q$ by an order of magnitude higher than that heuristically assumed
in Refs. [9, 46]. Namely, using rather thick wires of polished copper we may
reach quality factors higher than 3000, whereas further improvement may be
granted by cryogenic environment. We have validated the analytic model not
only by numerical simulations but also by building and measuring a $10\times
10$ wire array in a cubic metal box. Our numerical and experimental results
are in good agreement and pave the way for further studies of large,
controllable plasma haloscopes as advantageous setups for the search for dark
matter.

##  II Analytical model of the resonator

At the most basic level, a WM resonator consists of an array of wires, which
in this work we will consider to be encased inside a metallic cavity. Using a
cylindrical cavity with a square lattice of wires results in the variable
distance between the edge of the metamaterial and the wall. Depending on the
wall spacing and how the cavity is filled, there is a possibility of
eigenmodes being formed in the air gaps at frequencies comparable or lower
than the plasma frequency. We consider a cubic cavity as it matches the
lattice geometry, leading to a constant and simple distance between the cavity
walls and wires. The walls ensure that the only losses in the system are
either resistive losses or coupling to an external antenna, leading to a
stronger resonance. To understand the general behavior of such a system, we
will start with an analytic approach.

For simplicity we will take the WM to be a square grid of $N\times N$ metal
wires of thickness $2r$ arranged with a period $a$ (as depicted in Fig. 1). We
will consider wires with both rectangular and round cross-sections. Lateral
walls of the cubic metal cavity are distanced by $a/2$ from the centers of the
edge wires so that the side length $d$ of the resonator is equal to $d=Na$.
The distance of $a/2$ was chosen since it is the distance by which the
effective medium extends out of the material [50]. The wire length is defined
as $d$ and the wires are terminated at the top and bottom walls.

(a)

(b)

Figure 1: Left: The corner of the $xy$ cross-section for a resonator
consisting of an array of square wires of thickness $2r$, with spacing $a$
with a gap of $a/2$ between the wire array and an enclosing metal cavity of
side length $d$. Right: 3D model of the $10\times 10$ example wire medium
cubic resonator which will be used in numerical simulations. The wires have a
square cross-section for ease of calculation with a hexahedral mesh in CST MWS
[51] Eigenmode solver. The number of wires allows for manageable computational
requirements while operating in a regime where one expects the homogeneous
effective medium description to be valid.

The dielectric permittivity tensor $\hat{\varepsilon}$ of a simple WM composed
of $z$-oriented wires is given by (see, e.g., [2])

${}\hat{\varepsilon}=\left(\begin{matrix}1&0&0\\\ 0&1&0\\\
0&0&\varepsilon_{zz}\end{matrix}\right)\,,$ (1)

where in the absence of losses we have:

${}\varepsilon_{zz}=1-\frac{k_{p}^{2}}{k_{0}^{2}-k_{z}^{2}}\,,$ (2)

where $k_{0}$ is the free space momentum ($\omega/c$) and $k_{z}$ is the
momentum in the $z$ direction. This gives a maximally anisotropic plasma, so
that modes $E$-fields perpendicular to the wires "perceive" an almost empty
cavity. For the special case when the $E$-field is aligned strictly in the
$z$-direction (i.e., along the wires) this formula becomes identical to that
describing the permittivity of a non-magnetic collision-free plasma:

${}\varepsilon\equiv\left.\varepsilon_{zz}\right|_{k_{z}=0}=1-\left(\frac{\omega_{p}}{\omega}\right)^{2}.$
(3)

In writing these we have written the effective plasma frequency as
$\omega_{p}$ with $k_{p}=\omega_{p}/c$ the plasma wave number defined in [52]
for round wires as

${}k_{p}^{2}=\frac{2\pi/a^{2}}{\ln{\frac{a}{2\pi r}}+F(1)}\,.$ (4)

Here F is a function of the ratio of the periods in the lattice of wires for a
rectangular array. For square lattices $F(1)=0.5275$ [52]. As we only deal
with the case of square lattices, we will write $F(1)\equiv F$. In other
words, the effective plasma frequency and wavenumber are simply given by the
geometry of the system, depending only on $a$ and $r$. This allows for custom
plasmas to be created with the plasma frequency desired for any particular
application, or even tuned mechanically [9, 46].

### Modes

The eigenmodes of the whole structure (WM in the cavity) can be split into
Transverse Electric (TE) modes with the magnetic field aligned with the wires
and Transverse Magnetic (TM) modes with the electric field aligned instead.
For the TE modes, the $E$-field vectors lack any component along $z$ and if
the wires are thin enough their quasi-static interaction with the wires is
negligible. As a result, the frequencies of TE modes are nearly equal to the
frequencies of an empty cavity. Because of this, we will focus primarily on
the TM modes, however if the WM is extended to have wires in the
$x,y$-directions we would expect a similar story to play out for the TE modes.

For the TM modes, the $E$-field is confined to the $z$ axis and so sees a
plasma medium. Below the plasma frequency the effective refractive index is
predominantly imaginary and so any waves are evanescent. Near the plasma
frequency, the WM enters an ENZ regime leading to significantly larger
wavelengths. Because of this the TM modes are shifted to just above the plasma
frequency, leading to the fundamental mode being much higher in frequency than
would be realized for an empty cavity.

Neglecting losses, by requiring that the $E$-fields satisfy the conducting
boundary conditions at the walls, the TM modes $E_{z}^{lmn}$ with
eigenfrequencies $\omega_{\rm res}^{lmn}$ of amplitude $E_{0}$ can be written
as

$E_{z}^{lmn}=E_{0}\cos(k_{x}x)\cos(k_{y}y)\cos(k_{z}z)\,,$ (5)

with eigenfrequencies given by

$\epsilon\left(\frac{\omega_{\rm
res}^{lmn}}{c}\right)^{2}=\left(\frac{l\pi}{d}\right)^{2}+\left(\frac{m\pi}{d}\right)^{2}+\epsilon\left(\frac{n\pi}{d}\right)^{2}\,,$
(6)

where we have used that $k_{x}=l\pi/d,k_{y}=m\pi/d$, and $k_{z}=n\pi/d$. As
one of the primary applications of resonant WM are plasma haloscopes, we will
focus on the fundamental (TM110) mode. As this mode has the highest spatial
homogeneity, it would couple most strongly to dark matter and provide the
strongest signal [9]. In this case, we will define $\omega_{\rm
res}\equiv\omega_{\rm res}^{110}$, which allows us to write

$\left(\frac{\omega_{\rm res}}{c}\right)^{2}\equiv k_{\rm
res}^{2}=k_{p}^{2}+2\left(\frac{\pi}{d}\right)^{2}.$ (7)

We thus have a simple analytic expression for the resonant frequencies of the
resonant WM. To check this analytic effective medium treatment, we compare the
analytic calculations to full-wave simulations in CST Microwave Studio [51]
using the eigenmode solver in Fig. 2. By showing the resonant frequency versus
the number of wires across the resonator ($N=d/a$) we see that the analytic
formulas provide an excellent prediction for the full numerical treatment. The
agreement is particularly good for larger systems (1%), meaning that the
analytical treatment should be appropriate for the large sizes considered for
plasma haloscopes. We note, however, that this numerical model used a square
cross-section of wires for computational simplicity and faster meshing. As
will be shown later, modeling a cavity with round wires results in a small but
noticeable difference in the quality factor and plasma frequency. We see also
that as the size of the WM increases the $\omega_{\rm res}\to\omega_{p}$
asymptotically. With this understanding of the mode structure, we can turn our
attention to the losses.

Figure 2: Resonance frequency $f_{\rm res}$ as a function of the number of
wires $N^{2}$ in a WM consisting of radius $r=1$ mm wires with a period $a=1$
cm placed inside a cubic cavity. Analytic results (red line) assuming a
homogenization model are compared with a numerical simulation (blue squares)
in CST [51] with square wires of thickness $2r$, with a gap of $a/2$ with the
cavity walls. For comparison, the plasma frequency of the WM is shown via the
black dashed line. As $N$ increases $f_{\rm res}$ approaches the plasma
frequency of the WM.

### Losses

Unfortunately, normal conductors will always have resistive losses, leading us
to one of the most salient properties of any resonator: the quality factor
$Q$. To estimate the losses of the system, we must augment the dielectric
constant to include an imaginary component. For simplicity, we will consider
the same fundamental TM110 mode; however, other modes of the system can be
analyzed similarly. In this case, we can treat the medium as isotropic and use
the results of Ref. [53] to write

$\varepsilon=\varepsilon^{\prime}-j\varepsilon^{\prime\prime}=1-\frac{k_{p}^{2}}{k^{2}-j\xi
k}\,,$ (8)

with real and imaginary parts given by

$\displaystyle\varepsilon^{\prime}$
$\displaystyle=1-\frac{k_{p}^{2}}{k^{2}-\xi^{2}}\,,$ (9)
$\displaystyle\varepsilon^{\prime\prime}$ $\displaystyle=\frac{\xi
k_{p}^{2}/k}{k^{2}-\xi^{2}}\,.$

The loss coefficient $\xi$ is derived in Ref. [53]

${}\xi=\frac{Z_{w}}{L_{w}}\sqrt{\varepsilon_{0}\mu_{0}}\,,\\\ $ (10)

with $Z_{w}$ and $L_{w}$ being the wire impedance and inductance per unit
length, respectively. For wires with circular cross-sections, these quantities
can be calculated from the magnetic permeability $\mu$ and conductance
$\sigma$ of the wire material as [53, 54]

$\displaystyle L_{w}$ $\displaystyle=\frac{\mu_{0}}{2\pi}({\ln{\frac{a}{2\pi
r}}+F})\,,$ (11a) $\displaystyle Z_{w}$
$\displaystyle=\text{Re}\left(\frac{\sqrt{-j\omega\mu}}{\sqrt{\sigma}2\pi
r}\frac{J_{0}(\sqrt{-j\omega\mu\sigma}r)}{J_{1}(\sqrt{-j\omega\mu\sigma}r)}\right)\,.$
(11b)

The electromagnetic field of the fundamental mode of a cubic resonator
($d\times d\times d$), loaded by a dielectric whose complex permittivity has
real part $\varepsilon^{\prime}$ much larger than
$\varepsilon^{\prime\prime}$, can be obtained from Eq. (5) explicitly as

$\left.\begin{aligned}
H_{x}&=jE_{0}\sqrt{\frac{\varepsilon_{0}\varepsilon^{\prime}}{2\mu_{0}}}\cos{\frac{\pi
x}{d}}\sin{\frac{\pi y}{d}}\\\
H_{y}&=-jE_{0}\sqrt{\frac{\varepsilon_{0}\varepsilon^{\prime}}{2\mu_{0}}}\sin{\frac{\pi
x}{d}}\cos{\frac{\pi y}{d}}\\\ E_{z}&=E_{0}\cos{\frac{\pi x}{d}}\cos{\frac{\pi
y}{d}}\end{aligned}\right\\}-d/2\leq x,y\leq d/2\,,$ (12)

where we have assumed that the medium is non-magnetic ($\mu=\mu_{0}$). The
quality factor $Q$ of the resonator at the mode eigenfrequency can then be
found via the stored energy $W_{\text{stored}}$ and the power of the losses in
the walls, $P_{\text{walls}}$, and in the wires, $P_{\text{wires}}$, as
follows:

$Q=\frac{W_{\text{stored}}}{W_{\text{lost}}}=\frac{\omega
W_{\text{stored}}}{P_{\text{walls}}+P_{\text{wires}}}\,.$ (13)

The equation for the stored energy density in a dispersive dielectric includes
a correcting coefficient
$\frac{\partial(\omega\varepsilon^{\prime}(\omega))}{\partial\omega}$ and is
thus written as [55]

$\displaystyle w_{\text{stored}}$
$\displaystyle=\frac{\mu_{0}|H_{t}|^{2}}{4}+\frac{\varepsilon_{0}S|E_{z}|^{2}}{4}\,,$
(14) $\displaystyle S$
$\displaystyle=\frac{\partial(\omega\varepsilon^{\prime}(\omega))}{\partial\omega}=1+\left(\frac{\omega_{p}}{\omega_{\rm
res}}\right)^{2}\,.$

In the limit of $\omega_{\rm res}\xrightarrow{}\omega_{p}$ the coefficient
$S=2$ and the $H$-fields vanish. Integrating $w_{\rm stored}$ over the
resonator volume gives:

$W_{\text{stored}}=\iiint_{-d/2}^{d/2}w_{\rm
stored}dV=\frac{\varepsilon_{0}d^{3}E_{0}^{2}}{16}(\varepsilon{}^{\prime}+S)\,.$
(15)

Losses in the walls, $P_{\rm walls}$, can be found by integrating the
tangential components of the magnetic field over the walls. Ignoring an
overall phase and using that the currents are given by the cross product of
the normal vector $\bf\hat{n}$ via
${\mathbf{J}}=\mathbf{\hat{n}}\times\mathbf{H}$ we get

$\displaystyle\mathbf{J}\big{|}_{|x|=\pm d/2}$
$\displaystyle=E_{0}\sqrt{\frac{\varepsilon_{0}\varepsilon^{\prime}}{2\mu_{0}}}\cos{\frac{\pi
y}{d}}\mathbf{\hat{z}}\,,$ (16a) $\displaystyle\mathbf{J}\big{|}_{|y|=\pm
d/2}$
$\displaystyle=E_{0}\sqrt{\frac{\varepsilon_{0}\varepsilon^{\prime}}{2\mu_{0}}}\cos{\frac{\pi
x}{d}}\mathbf{\hat{z}}\,,$ (16b) $\displaystyle\mathbf{J}\big{|}_{|z|=\pm
d/2}$ $\displaystyle=\pm
E_{0}\sqrt{\frac{\varepsilon_{0}\varepsilon^{\prime}}{2\mu_{0}}}\sin{\frac{\pi
x}{d}}\cos{\frac{\pi y}{d}}\mathbf{\hat{x}}$
$\displaystyle\leavevmode\nobreak\ \,\,\,\,\,\pm
E_{0}\sqrt{\frac{\varepsilon_{0}\varepsilon^{\prime}}{2\mu_{0}}}\cos{\frac{\pi
x}{d}}\sin{\frac{\pi y}{d}}\mathbf{\hat{y}}\,,$ (16c)

we can then integrate to find [56]

$\displaystyle P_{\rm walls}$
$\displaystyle=\frac{R_{s}}{2}\iint_{-d/2}^{d/2}|\mathbf{J}|^{2}dS$ (17)
$\displaystyle=\frac{3R_{s}\varepsilon_{0}\varepsilon^{\prime}}{4\mu_{0}}d^{2}E_{0}^{2},$

where $R_{s}=\sqrt{\omega\mu_{0}/2\sigma}$ is the surface resistivity of the
cavity walls.

Figure 3: Unloaded quality factor $Q$ as a function of the number of wires
$N^{2}$ in a WM consisting of radius $r=1$ mm wires with a period $a=1$ cm
placed inside a cubic cavity. Analytic results (red line) assuming a
homogenization model are compared with a numerical simulation (blue squares)
in CST with square wires of thickness $2r$, with a gap of $a/2$ with the
cavity walls. For comparison, the quality factor of the resonator with
lossless walls $Q_{\rm wires}$ is shown (black dashed line). As $N$ increases,
$Q$ asymptotically approaches that of an infinite wire medium (i.e., no wall
losses) and $Q_{\rm wires}$ and $Q$ become virtually indistinguishable. In the
limit of $N\to 0$, $Q_{\rm wires}$ approaches infinity, as there are no wires
to cause resistive losses.

Similarly, the losses in the wires can be found by considering the imaginary
component of the dielectric constant $\varepsilon^{\prime\prime}$ using [55]

$P_{\rm
wires}=\frac{\omega\varepsilon{}_{0}\varepsilon{}^{\prime\prime}}{2}\iiint_{-d/2}^{d/2}|E_{z}|^{2}dV=\omega\frac{\varepsilon{}_{0}\varepsilon{}^{\prime\prime}d^{3}E_{0}^{2}}{8}\,.$
(18)

As opposed to the losses in the walls, which grow with the surface area of the
resonator, the losses in the wires grow proportionally to the cavity volume.
As a result, as $d$ is increased, $P_{\rm wires}$ dominate over $P_{\rm
walls}$ if the cavity is made of similar quality material. As axion
experiments will focus on large volumes (and so large $d$) the wall losses can
be mostly neglected. The resulting quality factor of the resonator can then be
written using Eqs. (15) and (18) as:

$Q_{\rm wires}=\frac{\omega W_{\rm stored}}{P_{\rm
wires}}=\frac{(\varepsilon{}^{\prime}+S)}{2\varepsilon{}^{\prime\prime}}\xrightarrow{k_{\rm
res}=k_{p}}\frac{k_{p}}{\xi}-\frac{3}{2}\frac{\xi}{k_{p}}\,.$ (19)

As can be seen from Fig. 3 for a large resonant cavity in the case of a highly
conducting metal ($\xi\ll k_{p}$), the quality factor depends on the size of
the resonator in a manner similar to that of the mode frequency. Our analytic
calculation is in excellent agreement with numerical simulations for all
system sizes. The worst agreement ($3^{2}$ wires with 2.4%) provides a good
match between analytics and simulations, improving to less than 1% for
$12^{2}$ and up. However, we can see that as the system size increases, the
relative importance of wall losses decreases, with $Q\simeq Q_{\rm wires}$ for
arrays larger than $\sim 7\times 7$, approaching an asymptotic value of
$Q\simeq 3300$. Beyond this point, increasing the system size simply gives an
almost linear increase in signal power, which is proportional to $QV$. These
analytical calculations were confirmed by full-wave simulations of square
cross-section wires in CST MWS [51] Eigenmode Solver, whose results are shown
in Fig.3 as well. As the number of wires approaches the limit of a single wire
$N=1$ both resonance frequency and Q-factor rise sharply, approaching the
corresponding values calculated for an empty cubic metallic cavity with side
length $d=a$. This can be explained by the losses in the walls starting to
dominate over the losses in the medium in smaller WM-filled cavities.

While the quality factor remains constant for large system sizes, the
existence of a single cavity mode actually breaks down. To see this, consider
the refractive index $n=\sqrt{\varepsilon\mu}$ near resonance, as shown in
Fig. 4. Near the plasma frequency the real and imaginary part of $n$ are equal
up to a sign which depends on the time convention used (i.e., $e^{\pm i\omega
t}$) with $\varepsilon\simeq j/Q_{\rm wires}$ leading waves to decay with a
characteristic length $\sim\sqrt{Q_{\rm wires}}\lambda_{c}$. This affects both
the energy transport in the system, and the ability to form standing wave
cavity modes. For cavities larger than this typical scale, waves will decay
before they travel from one wall to another. In this case, our mode analysis
breaks down. However, the analysis of Ref. [9] does not assume that cavity
modes form, and is thus unaffected. The axion would create an effective volume
current, exciting the entire system uniformly. However, as the system would
then be larger than the decay length, multiple antennas would be required to
pick up the full signal. While we leave such considerations for future work,
we note that the walls can be neglected and the quality factor of the cavity
will only depend on the power extracted by the antenna and the resistive
losses of the wires, which will still be given by $Q_{\rm wires}$.

Figure 4: Real (green) and imaginary (blue) parts of the refractive index $n$
as a function of frequency $f$ around the plasma frequency $f_{p}$. For
illustrative purposes, we have chosen a plasma with $\xi=10^{-3}k_{p}$, i.e.,
a lifetime of 1000 cycles.

Many applications of resonators rely on the resonance being high quality. With
this in mind, how can one maximize the quality factor? In general, the losses
will depend primarily on the thickness of the wires, as well as their
conductivity. While one might anticipate that thicker wires would result in
lower resistive losses in analogy to energy transport in power lines, this
does not hold without limit. As the radius of the wires increases it starts to
become comparable to the Compton wavelength, violating the assumption that the
wire radius is much smaller than the wavelength (i.e., that the wires are
thin).

As shown above, for large resonators the wall losses can be neglected, so for
simplicity we will treat the system as infinitely large (i.e., $f_{\rm
res}=f_{p}$ and $Q=Q_{\rm wires}$). This allows us to choose an optimal wire
radius for a given plasma frequency. To see how $Q$ scales with experimental
parameters, we can explicitly write it as

$Q\simeq\frac{k_{p}}{\xi}=\frac{\sqrt{\omega_{p}\sigma\mu^{-1}}\mu_{0}r}{{\rm
Re}\left[\sqrt{-j}\frac{J_{0}(\sqrt{-j\omega_{p}\mu\sigma}r)}{J_{1}(\sqrt{-j\omega_{p}\mu\sigma}r)}\right]}\left(\ln{\frac{a}{2\pi
r}}+F\right)\,.$ (20)

As we expect that larger wires will result in lower losses, we will focus on
the regime where $r\gg 1/\sqrt{\omega\sigma\mu}$. In this limit the wires are
much larger than the skin depth of the material
$\delta=\sqrt{2/\omega\sigma\mu}$ and we can simplify Eq. (20) by noting that

$\frac{J_{0}(\sqrt{-j}\alpha)}{J_{1}(\sqrt{-j}\alpha)}\lim_{\alpha\to\infty}=j\,,$
(21)

giving

$Q\simeq\mu_{0}r\sqrt{\frac{2\omega_{p}\sigma}{\mu}}\left(\ln{\frac{a}{2\pi
r}}+F\right)=2\frac{\mu_{0}}{\mu}\frac{r}{\delta}\left(\ln{\frac{a}{2\pi
r}}+F\right)\,.$ (22)

We see that $Q$ is determined by two factors. The first being given by the
ratio of radius $r$ with the skin depth $\delta$ (up to the relative
permeability, which for copper is close to unity). The second is a geometric
term coming from the inductance $L_{w}$, which regulates $Q$ for high
frequencies. Note that $a$ is uniquely determined for a given $r$ and
$\omega_{p}$, falling with increasing $\omega_{p}$ and increasing with larger
$r$. We can rearrange Eq. (4) for the explicit form, given by

$a=\frac{2\sqrt{\pi}}{\omega_{p}}{W}_{0}\left(\frac{e^{2F}}{\pi
r^{2}\omega_{p}^{2}}\right)^{-1}\,,$ (23)

where ${W}_{0}(x)$ is the principle branch Lambert $W$ function.

For some specific examples, we show $r=1,3\,$mm examples with $Q$ a function
of $f_{p}=\omega_{p}/2\pi$ for room temperature copper in Fig. 5. Due to the
trade-off between decreasing the resistive losses and maintaining a thin wire
limit, each wire radius has a maximum quality factor at a frequency which
corresponds to $r\simeq\lambda_{c}/50$. However, the peak of the quality
factor curve of a specific radius does not actually give the optimal wire
radius for a specific frequency.

To find an optimal wire radius, we maximize the quality factor $Q$ for a given
plasma frequency $f_{p}$, shown by the dashed line in Fig. 5. We see that the
maximal $Q$ falls as $1/\sqrt{f_{p}}$ following the skin depth, assuming that
the conductance is constant. Numerically, the maximum quality factor for a
given frequency occurs when $r\simeq\lambda_{c}/11$. As this trade-off is
caused by the geometry of the system, the optimal thickness of the wires is
largely unaffected by the conductance of the wires, which only modifies the
extrema of $Q$ through the Bessel functions in Eq. (20). Note that as the wire
thickness starts to become an appreciable fraction of $\lambda_{c}$, the
assumption that they are one-dimensional objects breaks down, which would
necessitate modifying the effective medium approach. Thus while these results
are indicative of the $Q$ that can be achieved, a practical design should be
fine-tuned through numerical simulation.

Figure 5: Unloaded quality factor $Q$ as a function of the plasma frequency
$f_{p}$ for an infinite copper wire metamaterial at room temperature. We show
wires with radius $r=1$ mm and $r=3$ mm in blue and green, respectively, with
the black dashed line showing the maximum possible $Q$ for a given $f_{p}$.
The period of the wires $a$ is adjusted so that the plasma frequency of the
system is given by $f_{p}$. The theoretically maximum $Q$ diverges as one
moves to DC as the optimal wire size and spacing approaches infinity.

To give a sense of scale, Refs. [9, 46] assumed $Q=100$ over a range of
$2-100\,{\rm GHz}$ in a cryogenic environment. However, as we have shown even
with room temperature copper, it is possible to get $Q>10^{3}$ over this
entire range, a gain of more than an order of magnitude. In a cryogenic
environment we would expect further gains, ultimately being limited by the
anomalous skin effect. For an example, a cavity designed specifically to
search for axion dark matter was demonstrated to improve in quality factor by
a factor of $\sim 3$ at 12 GHz by moving to cryogenic temperatures [57].
Further gains could be made by moving to superconducting wires, though this
may present design challenges in strong magnetic fields. We have also
demonstrated that with some simple optimization one can noticeably improve the
quality factor for a given frequency. This allows for high-quality WM
resonators to be manufactured, with particularly interesting applications in
the low-frequency regime.

##  III Resonator eigenmode simulations

While analytic formulas provide great simplicity and transparency, they will
necessarily neglect the fine details of the system. Further, the region of
validity of such a homogenized effective medium approach must be explored.
Testing the validity of the analytic approach requires a comparison with both
simulations and physical measurements. To accomplish the former, we use the
Frequency Domain Solver in CST MWS 2020.

To look at the mode structure directly, we will take the case of a $10\times
10$ wire array, using $r=1$ mm copper wires with a period of $a=1$ mm
(arranged as shown in Fig. 1). We compare the full wire treatment with a
completely homogeneous effective medium, with properties as calculated in Sec.
II. To start, we plot the $E$ and $H$-field distributions of the TM110 mode in
Fig. 6. Comparing the two red colour maps showing the value of the $E$-field
we see that the presence of the individual wires does not have a significant
impact on the overall mode structure for the $E$-field, except a local
decrease in the immediate proximity of the wires.

However, the sharp decrease in $E$-field around the wires causes significant
$H$-fields to be generated. Because of this the magnetic field circles around
individual wires, and the highest amplitudes are achieved not at the centers
of the walls, but at the cavity center (same as for the $E$-field). Indeed, it
is the mutual inductance of the wires that is responsible for the collective
oscillation of WM [1]. While this change does not have an impact for the
unloaded $Q$, it may be significant when we consider the coupling of external
circuits to the resonator. An empty cavity would effectively couple to a
magnetic probe inserted at the internal side of the wall. In a wire-filled
cavity, however, the strong local $H$-fields near the wires may affect the
loop antenna very differently. For instance, since the magnetic fields of any
two neighbouring wires to an extent counteract each other at the point between
them, placing a loop antenna there would lead to a significantly weaker
signal. Note that the $H$-fields around the wires are approximately three
times stronger than the strongest $H$-fields in the homogenized effective
medium approach. Therefore SMA (sub-miniature A-version) ports with an 8-mm-
long extended core placed at the point of the maximum of the $E$-field on the
top and/or bottom walls were used as monopole antenna probes in both
simulations and experiments.

(a)

(b)

Figure 6: The TM110 mode structure in the $xy$ cross section of a WM loaded
cavity calculated in a CST MWS Eigenmode solver. The amplitudes of the
$E$-field scaled via its maximum value ($E_{\rm max}$), $E/E_{\rm max}$, are
shown via the colour map (red). It is clear that the distribution of the
electric field is almost the same in two plots and corresponds to the one
given by Eq. (12). The log-scaled in-plane components of the $H$-field are
shown via the blue arrows. The direction (amplitude) of the $H$-field
corresponds to the direction (magnitude) of the arrows. Left: A full
simulation with metal wires. The WM consists of a $10\times 10$ array of
square wires of thickness $2r$, with spacing $a=1$ cm and $r=1$ mm inside a
metal cavity as shown in Fig. 1. Right: A simulation with an effective medium
using the analytic formula in Ref. [52] using the same parameters, except
replacing the square wires of the numerical simulation with radius 1 mm.

In order to couple to the resonant cavity, two SMA ports were added at the
centers of the walls at which the wires were terminated. Being placed at the
point of the maximum electric field at these walls, the SMA probes are very
strongly coupled to the mode field making the resonator heavily loaded by
these lumped ports. The vertical position of the probes allows us to only
excite the TM modes of the resonator, without coupling to the numerous TE
modes of the cavity, which are weakly influenced by the wires. As can be seen
from the comparison in Fig. 7, in terms of field magnitude the presence of
probes changes the field structure in their vicinity, but does little in terms
of affecting the mode structure at large. However, a slight non-homogeneity is
added in terms of the phase of the dominating $E_{z}$ component.

Thus we can see that while the effective medium approach allows for a good
prediction for the overall properties of a WM resonator, the presence of wires
leads to local distortions which may have an impact on the coupling of
antennas to the system.

(a)

(b)

Figure 7: Cross section of the $E$-field mode structure of the TM110 mode of a
WM loaded resonator through the $x,z$-directions found with the CST MWS
Eigenmode solver. The WM consists of a $10\times 10$ array of square wires of
thickness $2r$, with spacing $a=1$ cm and $r=1$ mm inside a metal cavity as
shown in Fig. 1. Darker red indicates a stronger $E$-field. Left: TM110 mode
of an isolated resonator (i.e., no antenna). Right: TM110 mode of the
resonator with two SMA ports acting as 8-mm-long monopole probes added at the
center of the top and bottom walls. The configuration mirrors the one used in
the experiment.

##  IV Experiment

Lastly, analytic and numerical solutions must always be validated by actual
physical measurements. To this end, a brass prototype was built with the aim
of testing whether a $Q$-factor on the order of thousands could be achieved at
room temperature by enclosing the wires within a cavity. Yellow brass wires
(65% copper, 35% zinc) of $r=1$ mm were placed and welded within a $10\times
10\times 10$ cm brass cube with a period $a=1$ cm. The lateral walls were
spaced $a/2=0.5$ cm away from the WM sample. To facilitate measurements, two
SMA ports were attached to the top and bottom walls to which the wires were
welded, as shown in Fig. 8. The two ports acted as 8-mm long monopole probes
coupled to the TM modes of the resonator. The center contact of the ports had
a diameter of 1.3 mm and was made of beryllium copper. The insulator
dielectric had a diameter of 4.1 mm and was made of polytetrafluoroethylene
(PTFE). The conductivity of the metal used in our experiment was measured in
the DC regime to be $1.51\times 10^{7}$ S/m. This conductivity is only
slightly lower than the tabulated value $1.59\times 10^{7}$ S/m of the 65%
brass that was used in the CST model for the numerical comparisons.

(a)

(b)

Figure 8: Photos of the experimental prototype WM resonator. The resonator is
a 10$\times$10 array of radius $1$ mm circular cross-section brass wires
placed with a period $a=1$ cm. The wires are inserted into the holes in the
brass walls and soldered to ensure an electric connection. Two 8-mm-long SMA
connectors acting as monopole antenna probes are inserted at the centers of
the walls to which the wires are connected. Left: The wires inside the
resonator in an unfinished state. Right: The finished resonator fully enclosed
in a cavity.

To measure the system, we used an Agilent E8362C vector network analyzer (VNA)
in the range of 11-13.5 GHz connected to SMA connectors acting as monopole
antennas. We first measured the coupling coefficient of each of the ports by
attaching the other port to a matched load and measuring the reflection
coefficient $S_{11}$ at the port in question. After employing curve fitting to
the resulting $S$-parameters to account for the coupling losses at the ports,
as described in Ref. [58], we obtained a pair of coupling coefficients
$\kappa_{m1},\kappa_{m2}$ for the matched setup.

We can convert the coupling coefficients of a matched cavity to input and
output coupling coefficients $\kappa_{1},\kappa_{2}$ via [58]

$\displaystyle\kappa_{1}$
$\displaystyle=\kappa_{m1}\frac{1+\kappa_{m2}}{1-\kappa_{m1}\kappa_{m2}}\,,$
(24a) $\displaystyle\kappa_{2}$
$\displaystyle=\kappa_{m2}\frac{1+\kappa_{m1}}{1-\kappa_{m1}\kappa_{m2}}\,.$
(24b)

The overall coupling coefficient can then be found as the sum of the input and
output ones,

$\kappa=\kappa_{1}+\kappa_{2}\,.$ (25)

Thus armed with the coupling coefficient, we used a transmission-type
measurement to study the $Q$-factor of the system. In Fig. 9 we compare the
numerical simulation with the S12 measurement. Apart from the lowest TM110
mode around 11.45 GHz, two other modes - TM111 and TM112 can also be seen at
higher frequencies. While there is a good overall agreement, there is a
discrepancy of about 33 MHz in the frequency of the resonances. This
discrepancy is likely caused by the finite accuracy available for a given mesh
and can likely be neutralized by increasing its density. The loaded $Q$-factor
$Q_{L}$, corresponding to the full width at half maximum, can be read off from
the 3-dB width of the $S_{12}$ maximum corresponding to the mode in question.
To translate this reading into an unloaded $Q$-factor $Q_{U}$ we can use the
coupling coefficient via

$Q_{U}=Q_{L}(1+\kappa)\,.$ (26)

Figure 9: S12 (transmission) parameters for a WM consisting of a $10\times 10$
wire array with spacing $a=1$ cm inside a metal cavity. We depict both a CST
simulation (red line) similar to that shown in Fig. 1 and the measured values
of the prototype experiment (blue dashed line) as shown in Fig 8. The CST
simulation assumes 65% brass wires with radius $1$ mm, whereas the experiment
uses wires with the same radius but has a measured DC conductivity of
$1.51\times 10^{7}$ S/m (slightly lower than 65% brass). Circular cross-
section wires are used in both the experiment and the simulation. The $xz$
cross-sections of the corresponding TM modes are shown as insets next to the
resonances in the CST simulation.

We summarize the key parameters ($\kappa,Q,f_{\rm res}$, etc.) in Table 1. The
most significant difference occurs in the unloaded quality factor $Q_{U}$: the
experimental prototype is noticeably more lossy than the simulated one, that
results in wider bandwidth and lower $Q$. The difference in the coupling
coefficients might be attributed to the non-ideal electrical connection of the
port and leakage of energy in the experimental case resulting in lower
coupling.

The extra losses in the experimental setup may be attributed to a number of
factors: the presence of an oxidized layer on the wires (while it does not
influence the DC conductivity averaged over the wire cross section, the skin
effect means that the majority of the current occurs in the periphery of the
wire), the sub-millimeter roughness of the wires surface, the presence of the
drops of solder inside the resonator, and possible misalignment of the wires.
Using the expressions from our analytical model, these various losses can be
all factored into an effective conductivity $\sigma_{\text{eff}}$, which is
decreased compared to the measured one (already lower than the tabulated one)
and turns out to be equal to $0.77\times 10^{7}$ S/m, i.e., about twice lower
than the actual conductivity of brass. In Fig. 10 we present the dependence of
the unloaded quality factor on the conductivity $\sigma$ calculated
analytically for several metals in comparison with the $\sigma_{\text{eff}}$
observed in the experiment to better visualise the discrepancy and the extent
to which the quality factor may be improved with more refined prototype and
measurement.

| Experimental | Numerical
---|---|---
Frequency [GHz] | 11.420 | 11.453
Bandwidth [GHz] | 0.022 | 0.016
Loaded Q | 509 | 735
Coupling coefficient | 1.34 | 1.82
Unloaded Q | 1194 | 2074
Table 1: Comparison between numerical simulations in CST and measurements of
the experimental prototype for the key parameters of the TM110 mode. The WM
consists of a $10\times 10$ wire array with spacing $a=1$ cm inside a metal
cavity. The CST simulation assumes 65% brass wires with radius $1$ mm, whereas
the experiment uses wires with the same radius but has a measured DC
conductivity of $1.51\times 10^{7}$ S/m (slightly lower than 65% brass).
Circular cross-section wires are used in both the experiment and the
simulation. Figure 10: Dependence of unloaded quality factor $Q$ on the
conductivity of the wires $\sigma$ for an infinite wire media. The other
parameters of the resonator are the same as for Fig. 3. The three blue points
mark maximum unloaded quality factors achievable with wires made of common
metals, 65% brass, 91% brass, and pure copper. The gray point shows the
quality factor observed in the experiment and an equivalent conductivity that
would result in the same value for the otherwise ideal conditions.

While the resulting value of the unloaded Q-factor is 40% lower than expected,
the value obtained still speaks in favour of the quality factors on the order
of thousands being achievable with high-quality copper wires (and
manufacturing). Further investigations explore the sources of increased losses
to allow for the highest possible quality factors.

##  V Conclusions

In this work we have analyzed the behavior of a cubic microwave resonator
filled with uniaxial wire medium analytically, numerically, and
experimentally. One particularly interesting use of such a system is to build
a plasma haloscope [9]; however, previous work made no computation of the
quality factor of such a system, and assumed a homogenized effective medium
model. When a WM is enclosed inside a metal cavity, we find that the primary
source of loss comes from the conductivity of the wires themselves, with wall
losses only playing a role for small systems. While the quality factor
decreases as the size of the system increases, it plateaus at the value
expected from wire losses alone. Thus the system size can be increased to
allow for large volume (and thus high signal power) haloscopes without
negatively impacting the resulting quality factors. We also discussed the
limits in which the system can be treated as a single-mode resonator: when the
system size is larger than $\sim\sqrt{Q}\lambda_{c}$ such a description breaks
down. While a plasma haloscope can operate in such a limit, the decay length
in media would require a multi-antenna readout design (such as a phase-matched
array).

We performed numerical simulations in CST, which showed a very strong
agreement with the analytic model. This indicates that additional boundary
conditions are not needed for the case of a wire medium with an $a/2$ spacing
between the wires and the walls, which validates the homogenization approach.
These numerical simulations were then compared to an experimental prototype,
which showed good agreement in the mode structure and loaded quality factors.
However, the resulting unloaded quality factor was 40% lower than expected,
most likely due to flaws in the manufacturing of the prototype. Promisingly,
even with room temperature brass, a quality factor larger than a thousand was
easily obtained, more than an order of magnitude higher than assumed in Ref.
[9]. With high quality copper, machining and moving to a cryogenic environment
we expect that as much as an additional order of magnitude can be gained.

We have studied in detail for the first time the behaviour and quality factor
of WM loaded resonators, finding that they are extremely promising for the
purpose of detecting axion and dark photon dark matter.

## Acknowledgements

The authors thank Maxim Gorlach, Jón Gudmundsson, Grigorij Karsakov, Tove
Klaesson, Eugine Koreshin, Mathew Lawson, Ivan Matchenya, and Karl van Bibber
for helpful discussions and also thank the members of the ALPHA Consortium for
discussion and support. R.B. thanks Gréta Horváthová for her continued
support. A.J.M. is supported by the European Research Council under Grant No.
742104 and by the Swedish Research Council (VR) under Dnr 2019-02337
“Detecting Axion Dark Matter In The Sky And In The Lab (AxionDM)”. R.B. and
P.B. were supported by a grant for scientific school НШ-2359.2022.4 and
Priority 2030 Federal Academic Leadership Program.

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