S

Smith Alcock Davidson Emmins Hiller Bardsley Holloway Malfois Marshall Pizzey
Rogers Shebanova Snow Sutter Williams [a]N.
<EMAIL_ADDRESS>if different from

[a]Diamond Light Source Ltd, Diamond House, Harwell Science and Innovation
Campus, Didcot, Oxfordshire, OX11 0DE, United Kingdom [b]ISIS Neutron and Muon
Source, Science and Technology Facilities Council, Rutherford Appleton
Laboratory, Didcot, Oxfordshire, OX11 0QX, United Kingdom [c]ALBA Synchrotron,
Carrer de la Llum 2-26, 08290 Cerdanyola del Vallès, Barcelona, Spain
[d]King’s College London, Guy’s Campus, Great Maze Pond, London SE1 1UL United
Kingdom

# I22: SAXS/WAXS beamline at Diamond Light Source - an overview of 10 years
operation

A. J  S. G  L. S  J. H  J. C  P  M  A. R  C. L  S. E  O  T  J. P  E. P

###### Abstract

Beamline I22 at Diamond Light Source is dedicated to the study of soft matter
systems from both biological and materials science. The beamline can operate
in the range $3.7\text{\,}\mathrm{keV}22\text{\,}\mathrm{keV}$ for
transmission SAXS and $14\text{\,}\mathrm{keV}20\text{\,}\mathrm{keV}$ for
microfocus SAXS with beamsizes 240 x 60 ${\mathrm{\SIUnitSymbolMicro m}}^{2}$
spot [Full width half maximum (FWHM) Horizontal (H) x Vertical (V)] at sample
for the main beamline, and approximately 10 x 10 ${\mathrm{\SIUnitSymbolMicro
m}}^{2}$ for the dedicated microfocussing platform. There is a versatile
sample platform for accommodating a range of facilities, and user developed,
sample environments. The high brilliance of the insertion device source on I22
allows structural investigation of materials under extreme environments (for
example, fluid flow at high pressures and temperatures). I22 provides reliable
access to millisecond data acquisition timescales, essential to understanding
kinetic processes such as early folding events in proteins or structural
evolution in polymers and colloids.

Beamline I22, a versatile SAXS/WAXS beamline at Diamond Light Source is
presented, along with an overview of 10 years operation of the beamline.

##  1 Introduction

Small Angle X-ray Scattering (SAXS) provides essential information on the
structure and dynamics of large molecular assemblies in low order
environments. These are characteristic of living organisms and also many
complex materials such as polymers and colloids. Relevant active research in
the UK encompasses the fields of medicine [Ma2016, burton2019,
coudrillier2016, Al-Jaibaji2018, kudsiova2019], biology [Troilo2016,
McGeehan2011, Arnold2011, Salamah2018], the environment [Neill2018,
Seddon2016] and materials [Summerton2018a, Wychowaniec2018, Burton2017], and
includes studies of supramolecular organization in biomechanical systems
[Xi2018, Kampourakis2018, Sui2014a], corneal transparency [Morgan2018,
Hayes2017], biological membranes [Barriga2016, Slatter2018, Tang2014], polymer
processing [Stasiak2015, Wan2018, Heeley2013, Toolan2017], colloids
[Calabrese2018, Poulos2016, Mable2016], inorganic aggregates [Raine2018,
Bennett2015, Zhou2018], liquid crystals [Hallett2014, Prehm2018, Lehman2018],
and devices [Xia2017, Barrows2016].

##  2 Beamline overview

The scientific and technological challenges confronted by this diverse
community required a high resolution, high brightness, synchrotron beamline.
The first small angle scattering beamline at Diamond, I22, uses an in-vacuum
undulator source to deliver a high photon flux into a focused 240 x 60
${\mathrm{\SIUnitSymbolMicro m}}^{2}$ spot [Full width half maximum (FWHM)
Horizontal (H) x Vertical (V)] at sample for the main beamline, and
approximately 10 x 10 ${\mathrm{\SIUnitSymbolMicro m}}^{2}$ for a dedicated
microfocussing platform. This platform will the subject of a further technical
paper, and will not be discussed in detail here. The main beamline has the
potential for a continuous energy range of
$3.7\text{\,}\mathrm{keV}22\text{\,}\mathrm{keV}$ with, at present, a reduced
range of $14\text{\,}\mathrm{keV}20\text{\,}\mathrm{keV}$ for the microfocus
option. It is currently operated from
$7\text{\,}\mathrm{keV}20\text{\,}\mathrm{keV}$; lower energy operation would
require either helium or vacuum sample chambers. Depending on the energy used,
and the exact geometry of the scattering experiment, the q range achievable at
I22 is $0.0011\leq q(\textup{\AA}^{-1})\leq 9.45$. The SAXS camera is composed
of evacuated sections ($5\text{\times}{10}^{-5}\text{\,}\mathrm{mbar}$ of
flight tube with a nosecone incorporating the WAXS detector at the sample end
and ended by a 310mm diameter Kapton Window at the SAXS detector end). These
tubes can be connected, as required, to form camera lengths anywhere from 1.9
to 9.9m in 0.25m steps. The primary end station, with associated area
detectors for static and time resolved measurements, is capable of recording
the scattered radiation from samples contained in a range of commercial and
bespoke sample environments.

Figure 1: Layout of I22 Figure 1 index | Component | Distance from source (m)
---|---|---
1 | in-vacuum undulator 25mm period | 0.00
2 | customised aperture | 17.20
3 | white beam slits | 22.70
4 | double crystal monochromator (Si 111) | 24.00
5 a-d | monochromatic slits | 25.57, 28.55, 46.00, 47.00
6 | horizontal focussing mirror | 26.78
7 | vertical focussing mirror | 27.73
8 | sample position | $47.1\text{\,}48\text{\,}$
9 | wide angle detector | 0.17 (from sample)
10 | small angle detector |  $1.9\text{\,}9.9\text{\,}$ (from sample)
Table 1: Major components of I22

##  3 Undulator

The beamline operates a, Diamond designed [Patel2017], 2m in-vacuum undulator
with a period of 25 mm, providing continuous energy coverage over the energy
range of the beamline. The beamline is optimised for the energy range
$8\text{\,}\mathrm{keV}20\text{\,}\mathrm{keV}$ working primarily with the
5${}^{\text{th}}$ through to the 17${}^{\text{th}}$ harmonics. Off-axis
undulator radiation is removed to minimise heat load on the beamline optics,
via a water-cooled 150 x 75 $\mathrm{\SIUnitSymbolMicro rad}$ (H x V) aperture
located inside the storage ring tunnel at $17.202\text{\,}\mathrm{m}$
downstream from the source. Water-cooled primary slits further define the beam
before energy selection and focusing. The undulator has a very small phase
error ($0.02\text{\,}\mathrm{mm}$). Figure 2 shows the spectrum as a function
of energy and undulator gap. All available harmonics can clearly be
distinguished, from the 2${}^{\text{nd}}$ at the top left, through to he
20${}^{\text{th}}$ at the bottom right.

Figure 2: In-vacuum U25 undulator spectrum of I22. Flux values measured with
calibrated diagnostic after the monochromator. N.B. The primary White Beam
slits were closed to a central 100 x 100 ${\mathrm{\SIUnitSymbolMicro m}}^{2}$
gap to give clean undulator spectrum

##  4 Monochromator

The monochromator (Oxford Danfysik, Oxford, UK) is a fixed-exit double-crystal
design, originally established for 3 crystal sets. It is currently installed
with a Si(111) crystal pair; this allows access over the full operational
energy range, with a resolution of 1.6 x $10^{-4}$ $\Delta E/E$ at
$10.0\text{\,}\mathrm{keV}$. Vertical translation of the second crystal
maintains a fixed beam exit configuration for the beamline. Both the first and
second crystals are indirectly cooled by liquid nitrogen, to prevent damage
arising from the high power density of the undulator. An indium foil provides
thermal contact for both crystals to a cooled copper block. The first crystal
is cooled by liquid nitrogen flowing continuously through the copper block
while heat from the second crystal is removed via copper braiding. The Diamond
storage ring has run in top up mode since October 2008, with an injection
every ten minutes. This produces a very slight variation in ring current of
$\sim$2% or $\pm$$5\text{\,}\mathrm{mA}$ at most, therefore changes in the
heat load on the monochromator during an experiment are primarily due to
undulator gap and harmonic changes associated with a change in energy. Both
coarse motors ($\pm$$4\text{\,}\mathrm{mrad}$) and fine piezo actuators
($\pm$$180\text{\,}\mathrm{\SIUnitSymbolMicro rad}$) are used to adjust the
pitch and roll of the crystals, maintaining alignment of the first and second
crystal lattice planes.

###  4.1 Monochromator calibration

The monochromator is periodically calibrated using EXAFS spectra collected
from a series of retractable metal foils that are permanently installed in the
beamline downstream of the monochromator. Scatter diodes before and after the
foils record $I_{0}$ and $I_{t}$ values, respectively. EXAFS features from the
spectra (peaks), and the derivative of the spectra (edges) for all of the
foils (V, Fe, Cu, Zn, Au, Zr, and Mo) are fitted, against their expected
positions from literature values, [BEARDEN1967] in a global least squares
minimisation. The main part, lower left, of Figure 3 shows the overall fit.
The lower right section of Figure 3 shows the absorption spectra from V, Cu,
Au, and Mo as measured on I22. Expanded views of the linear fitting of the
vanadium, copper, gold, and molybdenum features are shown along the top,
demonstrating the consistency of the fit across a wide range of Bragg angles.
In a typical fit 46 data points, spanning $None$, are used to calibrate the
Bragg axis.

Figure 3: Calibration of the I22 monochromator using EXAFS from metal foils

As a way of evaluating the repeatability 10 repeat spectra were measured for
all of the above edges. The data is given in the supplementary information.
Standard deviations for the measured positions of EXAFS features of
$3.8\text{\,}\mathrm{\SIUnitSymbolMicro rad}$ at the vanadium edge, and
$1.8\text{\,}\mathrm{\SIUnitSymbolMicro rad}$ at the molybdenum edge were
found highlighting the excellent repeatability of the monochromator design.

##  5 Mirrors

The beamline focusing comprises a Kirkpatrick–Baez (KB) [Kirkpatrick1948]
mirror pair (ACCEL, Germany), using adaptive bimorph mirror technology,
[Susini1995, Cautero2007] to provide independent horizontal and vertical
focusing at any point in the end station while still operating close to 1:1
focusing. Bimorph mirrors have three advantages that make them useful for I22.
First, by varying the voltages on individual electrodes, one can correct the
waviness left on the mirror’s surface by the polishing process, thus achieving
a sharp focal spot with small tails. Second, a bimorph responds within seconds
to changes in voltage, and even settling times imposed by imperfect mounting
of the bimorph generally do not exceed 10-15 minutes. Therefore, bimorph
mirrors permit the size, shape and focal distance of the X-ray beam to be
changed rapidly to accommodate different camera lengths. Third, they have good
long-term stability, and thus can be left at a constant voltage setting over
weeks or months without loss of focal quality. The horizontal focusing mirror
($900\text{\,}\mathrm{mm}$ long, with an $820\text{\,}\mathrm{mm}$ active
length) and the vertical focusing mirror ($600\text{\,}\mathrm{mm}$ long, with
a $550\text{\,}\mathrm{mm}$ active length) are located at
$26.8\text{\,}\mathrm{m}$ and $27.7\text{\,}\mathrm{m}$ respectively from the
source and share a common vacuum vessel. Each mirror substrate is made of
fused silica, $50\text{\,}\mathrm{mm}$ wide with a central
$15\text{\,}\mathrm{mm}$ wide coated stripe of rhodium; the stripes provide
excellent harmonic rejection across the beamline’s energy range while operated
at $2.6\text{\,}\mathrm{mrad}$. The adaptive bimorph capability is provided by
piezo actuators attached to each substrate. The vertical mirror has 32
electrodes, which are paired along the length to give 16 virtual electrodes.
These are used to correct the mirror figure and provide vertical focusing. The
horizontal mirror has 12 electrodes, 4 at either end are coupled into 2 pairs
and the 4 remaining electrodes are left as singles to give finer control over
the central portion of the mirror, where it is of greatest benefit for
providing optimal horizontal focusing.

###  5.1 Mirror refurbishment

Examination of the Bimorph mirrors using the x-ray beam and the Diamond Light
Source Nanometre Optical Metrology (NOM) [Alcock2010] non-contact profiler
revealed damage at the piezo interfaces [Alcock2013] as indicated by the gross
changes in slope error across the mirror in the measurements shown in Figure
4. While not all junctions behaved the same (dotted lines in 4 show the
position of the junctions, and features are invariably found near to the
junctions), these effects translated into poor focusing performance rendering
optimum focusing impossible and a degradation in beamline background.
Repolishing the optical surface of both mirrors significantly improved the
focusing capability of the mirrors to the point where optimum focus is now
achievable. Data from the Diamond NOM shows a 9-fold improvement in slope
error after repolishing from $2.7\text{\,}\mathrm{\SIUnitSymbolMicro
rad}0.3\text{\,}\mathrm{\SIUnitSymbolMicro rad}$.

Figure 4: Effect of repolishing the I22 VFM as determined by Diamond NOM

We stress that the junction effect has not been found to appear in the latest
“second-generation” bimorph mirrors, which have their piezoelectric plates
attached to the sides of the mirror substrate rather than being sandwiched
between an upper and a lower substrate, as is the case with I22’s first-
generation bimorph mirrors[Alcock2019, Alcock2019a].

The table below gives an indication of the beamsize at various camera lengths
on I22 after correcting the mirror surfaces.

Focal Position |  Demagnified FWHM source size (HxV, $\mathrm{\SIUnitSymbolMicro m}$) |  H FWHM slope error broadening ($\mathrm{\SIUnitSymbolMicro m}$) |  V FWHM slope error broadening ($\mathrm{\SIUnitSymbolMicro m}$) |  Measured FWHM focal size (HxV, $\mathrm{\SIUnitSymbolMicro m}$)
---|---|---|---|---
Sample |  258 x 6 |  39 |  60 |  262 x 60
Sample + 3m |  296 x 7 |  45 |  69 |  299 x 69
Sample + 5m |  321 x 7 |  49 |  75 |  329 x 75
Sample + 7m |  346 x 8 |  52 |  81 |  350 x 81
Table 2: Beamsizes for the repolished Bimorph Mirrors

These measurements do not perfectly match the theoretical widths. However,
with reasonable values of electron beam size and mirror slope error, the
theoretical widths come within 16% of the measured horizontal widths, and 10%
of the measured vertical widths (Table 2).

##  6 Endstation

###  6.1 Flux calibration

Flux values are reported below for a selection of commonly used experimental
conditions. The measurements were taken using with a calibrated diode (Canbera
PD-300 PIPS diode calibrated by PTB Berlin) at the sample position.

Energy | Measured Flux
---|---
/ $\mathrm{k}\mathrm{e}\mathrm{V}$ | photons ${\mathrm{s}}^{-1}$
7 | $1.08\text{\times}{10}^{13}$
10 | $6.94\text{\times}{10}^{12}$
12.4 | $3.97\text{\times}{10}^{12}$
14 | $2.74\text{\times}{10}^{12}$
18 | $1.48\text{\times}{10}^{12}$
Table 3: Measured Flux for commonly used energies

###  6.2 Sample environments

Samples are mounted on a platform (IDT, Cheshire, UK; customised design)
allowing remotely controlled motion of $200\text{\,}\mathrm{mm}$ both
horizontally and vertically. The platform is mounted on rails for manually
controlled motion of $1000\text{\,}\mathrm{mm}$ along the beam path. Combined
with motorised X,Y and manual Z motion of the WAXS detector nosecone, any
reasonable size of sample environment can be accommodated whilst minimising
air gaps and attendant background scatter. The sample table surface comprises
a standard optical breadboard (750 x 750 ${\mathrm{mm}}^{2}$, M6 tapped holes
on a $25\text{\,}\mathrm{mm}$ pitch) which facilitates mounting of sample
environments, large and small. The platform has
$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ resolution, and can accommodate
large sample environments (up to $100\text{\,}\mathrm{kg}$), either provided
in house or built by users. If a small sample environment is required (up to
$2\text{\,}\mathrm{kg}$), such as a single sample cell or lightweight
capillary holder, 2 motorized translation stages equipped with stepper motors
(Newport Spectra-Physics Ltd, Didcot, UK; UTS100PP) provide precise sample
positioning with $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ resolution over a
$100\text{\,}\mathrm{mm}$ range of motion horizontally and vertically. The
usual sample environments found on versatile SAXS beamlines are also available
on I22. These can be controlled remotely, and trigger, or be triggered, by the
data acquisition software. Both Linkam DSC and Capillary furnaces (Linkam
Scientific, Surrey, UK) are available for use accessing temperatures in the
range
$-180\text{\,}\mathrm{\SIUnitSymbolCelsius}550\text{\,}\mathrm{\SIUnitSymbolCelsius}$.
Two commercial stopped flow apparatus (SM 400; Bio-Logic, Grenoble, France)
specifically designed for synchrotron radiation SAXS are available. One is
used for investigations of conformational changes in proteins [Panine2006],
nucleic acids and macromolecules, while the second one is dedicated to
material science studies [Lund2013]. A stress controlled rheometer (Physica
MCR 501; Anton Paar, Hertford, UK) is available with a range of geometries,
suitable for a wide range of viscosities ranging from water to polymer melts
[Poulin2016]. It can be temperature controlled using air or water, providing
an operating range between
$-20\text{\,}\mathrm{\SIUnitSymbolCelsius}350\text{\,}\mathrm{\SIUnitSymbolCelsius}$.
A micromechanical tensile tester has been developed in collaboration with
Queen Mary University of London [Karunaratne2012b]. This can be used in
tension, compression, and bending modes. A millisecond pressure-jump cell
($1\text{\,}\mathrm{bar}5000\text{\,}\mathrm{bar}$) for use in X-ray
scattering experiments has been developed in collaboration with Imperial
College London, and is now routinely available on I22 [Brooks2010]. For higher
pressure a Diamond Anvil cell is available for use with microfocus beam
(Almax- easyLab, Diksmuide, Belgium; Boehler-Almax PlateDAC). This cell covers
the pressure range $1\text{\,}\mathrm{GPa}60\text{\,}\mathrm{GPa}$.

###  6.3 Detectors

The beamline has a matched pair of Dectris (Dectris AG, Switzerland) Pilatus
P3 Hybrid silicon pixel detectors [Henrich2009], see Table 4. Of particular
note is the in vacuum L shaped WAXS detector. This has the basic form of a
Pilatus P3-2M with three modules in the lower right corner removed. The
detector is built into the vacuum space of the SAXS camera nosecone affording
partial 2D access to WAXS to complement full 2D SAXS in the appropriate
configuration. With careful beamline set up, these detectors allow overlap of
1D SAXS and WAXS data over the full camera length capability of the beamline
as highlighted by the data shown in figure 5. It is possible to arrange the
detectors such that no shadowing of the SAXS detector by the WAXS detector
occurs.

Figure 5: Overlapping q-scale of I22 detectors at 9m camera length, SAXS
q-scale is $0.0020.24$ ${\mathrm{\SIUnitSymbolAngstrom}}^{-1}$, WAXS q-scale
is $0.25.6$ ${\mathrm{\SIUnitSymbolAngstrom}}^{-1}$. Data are from
$100\text{\,}\mathrm{nm}$ silica spheres collected as a powder between 2
Scotch Tape${}^{\text{TM}}$ windows. NB: The WAXS data has been shifted upward
slightly to better show the overlap region Table 4: Detectors available on I22
and their main uses
Detector | Use
---|---
Pilatus P3 2M | (Gi)SAXS Data Collection
|  $172\text{\,}\mathrm{\SIUnitSymbolMicro m}$ pixel size
| Total active area $254\text{\,}\mathrm{mm}$ x $289\text{\,}\mathrm{mm}$
| Frame rate $250\text{\,}\mathrm{Hz}$ with $1\text{\,}\mathrm{ms}$ read out
Pilatus P3 2M-DLS-L | In vacuum (Gi)WAXS data collection
|  $172\text{\,}\mathrm{\SIUnitSymbolMicro m}$ pixel size
| Total active area $254\text{\,}\mathrm{mm}$ x $289\text{\,}\mathrm{mm}$
| Frame rate $250\text{\,}\mathrm{Hz}$ with $1\text{\,}\mathrm{ms}$ read out
Vortex Single Element | Fluorescence detection for mapping experiments
Silicon Pin Diode | Absorption correction data collection
|  $1\text{\,}{\mathrm{mm}}^{2}$ diode embedded in beamstop

##  7 Software

The beamline is controlled using an architecture based on the Experimental
Physics and Industrial Control System (EPICS) [Dalesio1994]. A graphical
synoptic of the whole beamline is available and windows for specific
components can be accessed to set and monitor process variables. Users
interact with the beamline through the generic data acquisition client
[Rees2010, Gibbons2011] (GDA) a software-based data acquisition system that
sits on top of the EPICS layer. For simple data collection, frames $\times$
collection length, there are a range of graphical user interfaces available
and for more complex experiments the full Jython scripting capability of GDA
can be employed.

After the acquisition of frame data, by default, GDA triggers any requested
analyses that are available through the Data Analysis WorkbeNch (DAWN)
software package [Basham2015, Filik2017]. Most commonly, this analysis entails
the reduction of frame data to a one-dimensional data set of intensity, I,
plotted against the scattering wave vector, q, following a standardised schema
[Pauw2017a]. It is also possible to analyse particular Debye-Scherrer rings,
producing I versus azimuthal angle, $\chi$ plots, the calculation of
orientation factors and degrees of orientation [Cinader1998, Hermans1944] as
well as crystallite thickness information from frame data [Fratzl1996]. These
scattering specific analyses are presented alongside a number of other common
analytical methodologies, such as function fitting, any number of which can be
automatically triggered following data acquisition.

##  8 Science examples

To illustrate the versatility and data quality of the beamline we represent
below a selection of science highlights from the 10 years of operation of I22.
These show the diversity of systems that can be studied on a flexibly designed
SAXS/WAXS beamline as well as illustrate the range of sample environments
available to users.

###  8.1 Solution Scattering

[OSullivan2011]

Large $\pi$-conjugated macrocycles are of interest for artificial
photosynthesis applications because of their resemblance to natural light-
harvesting complexes. Synthesis of these molecules is complicated by the need
to template, or support, the structure during synthesis. The synthesis of
larger target molecules requires commensurately larger scaffold molecules.
Eventually synthesis of the scaffold molecule can become as challenging as the
target molecule itself. Using the concept of a “Vernier complex”, a
combination of two molecular entities with an incommensurate number of
complementary bindings sites, researchers from Oxford and Nottingham
demonstrated that it is possible to build a porphyrin macrocycle in high
yield, in a facile manner from easily synthesised building blocks. The target
molecule was composed of 12 porphyrin units and synthesis was achieved from a
combination of a linear building block of 4 porphyrin units, along with a
circular hexapyridyl template. Two templates and three linear building blocks
spontaneously form into a “figure-of-eight” Vernier complex. After coupling of
the ends of the linear building blocks, the template can be removed to reveal
the cyclic 12-unit porphyrin molecule. Solution scattering data collected at
I22 confirmed the structures of both the “figure-of-eight”, and the free
cyclic molecule. At the time, the $4.7\text{\,}\mathrm{nm}$ diameter
12-porphyrin nanoring was among the largest $\pi$-conjugated macrocycles
synthesised. The group have continued to synthesise further related systems of
increasing size and complexity. [Sprafke2011, Cremers2018, Bols2018a]

###  8.2 Colloid Science

[Tang2014]

Cells are extremely complex systems and for life to form there are high
demands for a chemically rich, highly structured, environment in the centre of
the cell. Within the biological community there are, primarily, two competing
theories that predict the formation of protocells, widely regarded as
precursors to the cells we observe today and, therefore, a pre-requisite to
early life on Earth.

One thesis is based on fatty acids in water spontaneously forming a bilayer
membrane in water which, in turn, subsequently self-assemble into fatty acid
vesicles due to the partition function between the hydrophobic and hydrophilic
elements of the molecule and water[Deamer1982, Monnard2002]. The other is
derived from the formation of chemically rich liquid microdroplets forming
from omnipresent organic chemicals and natural polymers driving spontaneous
phase separation generating a rich mixture of coacervates in
solution[Oparin1953].

Combining these two theses, researchers from the University of Bristol Centre
for Protolife Research and the Chemistry Department of Imperial College London
have examined a hybrid protocell model based on the spontaneous self-assembly
of a fatty acid membrane on coacervate microdroplets.

Experiments conducted at I22 demonstrated that fatty acids assembled at the
surface of the coacervate droplets forming highly organised multilayers. No
multilamellar structures were seen in SAXS patterns at low fatty acid
concentrations. Despite the low contrast between the two systems, the use of
photon counting detectors on I22 meant that it was possible for Tang et al. to
determine the size and structure of the microcompartments, with some key
features of interest being observed between
$4\text{\,}\mathrm{nm}6\text{\,}\mathrm{nm}$ in size.

###  8.3 Environmental Science

[Bots2014]

There is a significant legacy of radioactive waste from nuclear power
generation and military activities. Currently the favoured route for disposal
of higher activity wastes is burial, deep underground, in a geological
disposal facility (GDF) [DEFRA2008] to contain the radionuclides until they
can decay into stable forms. Almost all designs for GDFs involve cement and
steel in their construction and backfill material. Once the facility is
closed, and becomes saturated with groundwater, interaction between water and
the cementitious material will generate a hyperalkaline leachate. Chemically
reducing conditions are also expected to prevail in the facility, since any
available oxygen will be locked up by corrosion of the steel components. The
most abundant radionuclide expected in such a GDF will be uranium which under
the conditions found in a GDF (high pH, slightly reducing) is most stable as
the U(VI) ion. U(VI) adsorbs strongly to many solid surfaces and this is
expected to limit the mobility of the radionuclide and help keep it within the
bounds of the GDF. A team of researchers from Manchester and Leeds found that
under the expected conditions of a GDF, uranium can form stable colloidal
nanoparticles that might be expected to have much higher mobility and
potentially escape from the GDF. Time-resolved SAXS data collected at I22
showed that these nanoparticles (1-2 nm in diameter) form within a few
minutes, under conditions relevant to a GDF, and coalesce into colloidally
stable aggregates within a few hours. The colloidal material remains stable
for over 2.5 years. These results show that GDF design will need to account
for the potential of uranium to form such nanoparticles and put in place
measures to mitigate their migration from the GDF.

###  8.4 Inorganic Chemistry

[Stawski2016]

Calcium sulfate is a well-known and commonplace mineral, with large deposits
of both the gypsum (CaSO4·2H2O) and anhydrite (CaSO4) crystalline phases found
naturally on Earth [Warren2006]. A third phase, bassanite (CaSO4·0.5H2O),
while of very limited abundance in natural settings, has an important role in
the construction industry, with 100 billion tones of it produced annually as
Plaster of Paris. Along with gypsum, bassanite has also recently been found as
a natural deposit on Mars. [Gendrin2005, Wray2010]

Despite its prevalence, the way in which gypsum forms from ions in solution
remains a mystery. Gaining a better understanding of the fundamental
nucleation and growth of the mineral is key if scientists want to control its
formation. Ultimately this could lead to more energy efficient routes of
making bassanite, a process which is currently very energy intensive.

Previous studies have used various Transmission Electron Microscopy approaches
to follow gypsum’s growth [Wang2012a, Driessche2012], however this technique
requires samples to be frozen or dried and studied under a vacuum, where each
process may introduce unintended artefacts. Mindful of this problem, a pan-
European team of researchers from institutes in the UK, Germany, Spain,
Belgium, France, and Denmark, used I22 to develop a method that was truly in
situ, and followed the nucleation and crystallisation in solution and in real
time.

Stawski et. al. found that the crystallisation proceeds via a four step
mechanism, which begins with the formation of “nano-bricks” - amorphous CaSO4
cores. These are well dispersed throughout the water matrix. In the second
step the bricks move slowly closer to each other, beginning to form unordered
domains and creating a disordered mosaic. Only in step three do the nanobricks
start to self-assemble into a wall like structure, with them crystallising
into gypsum in the fourth and final step.

###  8.5 Liquid Crystals

[Zeng2016]

Research into the formation and structure of liquid crystal systems has
delivered a myriad of technological and societal advances. Despite the
maturity of this field new exotic compositions and morphologies are
continually being discovered facilitating exciting possibilities in display
technologies, [Xiang2015, Wang2016] passive thermal stabilisation
[Ptasinski2014] and adaptive optics. [Arines2009, Harada2018] Concomitantly,
such studies have also extended our understanding of the mechanics within
cellular membranes [Lee2017] and the formation of certain biological
materials, such as silk. [Walker2015]

Advancing this domain further, researchers from the University of Sheffield
conducted a study revealing a novel liquid crystal structure where
bolaamphiphilic molecules were found to self-assemble into a $Pn\bar{3}m$
double diamond cubic liquid-crystalline phase. This work was the first
unambiguous demonstration of a thermotropic double diamond phase opening the
door to a family of new three-dimensional liquid crystal structured based
around the design principle of skeletal networks consisting of axial mesogen
bundles.

Measurements conducted on I22 were instrumental in the identification of this
new structural family via their morphological characterisation, in both the
SAXS and WAXS regimes, complementing previously obtained polarimetry and
differential scanning calorimetry measurements. The combination of the results
presented in these datasets together allowed Zeng et al. to conclusively
assign this phase, despite the unexpected structure uncovered.

###  8.6 Biomaterials

[Inamdar2017]

Articular cartilage is a natural biomaterial whose structure at the micro- and
nano-scale is critical for healthy joint function and where degeneration is
associated with widespread disorders such as osteoarthritis. At the nanoscale,
cartilage mechanical functionality is dependent on the collagen fibrils and
hydrated proteoglycans that form the extracellular matrix. [Setton1999,
Korhonen2002] The observation of the dynamic response of these ultrastructural
building blocks at the nanoscale has, however, proven elusive.

Inamdar et al. were able, for the first time, to observe the dynamics via
their utilisation of highly time-resolved, microfocussed, tensile stress and
strain SAXS measurements performed on I22. Using a micro-mechanical tester
[Karunaratne2012b] it was possible to observe the time-resolved changes in
collagen fibrillar D-periodicity during the compression of both bovine and
human cartilage explants.

Results obtained demonstrated the existence of a pre strain in the collagen
fibril, estimated from the D-period at approximately 1-2%. It is thought that
this is due to osmotic swelling pressure induced by the negatively charged
proteoglycans.

Uniquely, the data obtained shows a rapid reduction and recovery of this pre-
strain which occurs during stress relaxation, approximately 60 seconds after
the onset of peak load. Furthermore, the results highlight that this reduction
in pre-strain is linked to disordering in the intrafibrillar molecular
packing, alongside changes in the axial overlapping of tropocollagen molecules
within the fibril.

Inamdar et al. observed that tissue degradation in the form of selective
proteoglycan removal disrupts both the collagen fibril pre-strain and the
transient response during stress relaxation. This study bridges a fundamental
gap in the knowledge describing time-dependent changes in collagen pre-strain
and molecular organisation that occur during physiological loading of
articular cartilage.

This previously unknown transient response is likely to transform our
understanding of the role of collagen fibril nano-mechanics in the
biomechanics of cartilage and other hydrated soft tissues.

###  8.7 Nanostructures

[Brady2017]

There are a wide range of emerging technological applications [Rothemund2006,
Iinuma2014, Ke2012, Tian2015, Jones2015] that require materials with a
precisely defined 3 dimensional nanostructure. To achieve this the structures
must be built from the “bottom-up” and, since the molecular building blocks
are too small to manipulate directly, they must be encouraged to self-
assemble.

Many soft matter systems feature self-assembly but directing this towards
desired structures is problematic, particularly if the desired structure has
significant porosity. One class of materials that have the potential to
deliver bespoke porous 3D nanostructures are those made from self-assembled
DNA, dubbed “nanostars”. Despite their initial promise it has been found that
the DNA nanostructures suffer from a lack of flexibility that hinders their
ability to form contiguous crystalline networks. A team of researchers from
the University of Cambridge and Imperial College London proposed a
modification to the system that would relax the strict geometrical constraints
that prevent the DNA nanostructures from realising their potential.

By adding strands of cholesterol to the ends of the DNA strands they were able
to create a range of highly porous “C-stars”. It is the amphiphilic nature,
and variable binding valency of the cholesterol sections that give the C-stars
the required flexibility to form the intricate 3-d networks. Both bulk powder
SAXS and microfocus single crystal SAXS at I22 were used by the team to
characterise the structures. They continue to explore the diverse structures
that this technology has enabled.

##  9 Summary

I22 was designed as a multipurpose, undulator driven, combined SAXS-WAXS
facility to serve the needs of a broad UK community of SAXS/WAXS users. The
beamline was, therefore, designed as a very versatile platform capable of
accommodating a wide variety of biological and soft matter studies. The range
of science covered has grown during its operation to include users supported
by all of the major UK Science Research Councils as well as European funding
bodies and beyond. Being Diamond’s primary multipurpose SAXS beamline, it has
been required to carry out studies ranging from studying precursors to life on
earth [Tang2014], through self-assembly of polymer thin films [Pearson2014],
to microfocus SAXS mapping studies on bone [Karunaratne2013] and cornea for a
range of medical studies (e.g. [Hayes2013]). The selected examples illustrate
the potential of the I22 beamline as a high flux, high resolution, X-ray
scattering beamline for in situ and time resolved investigations.

Acknowledgements

We would like to thank the large number of Diamond Light Source personnel,
both past and present, that have been involved in the detailed design, build,
and commissioning of beamline I22 and its components. In particular we would
like to thank Alan Bone, Tom Cobb, Alan Day, Alan Grant, Andy Gundry, Ian
Johnson, Simon Lay, Pete Leicester, Emily Longhi, Ronaldo Mercado, Brian
Nutter, Mark Popkiss, Tobias Richter, Chris Roper, Irakli Sikharulidze, Andy
Smith, Richard Staunton-Lambert, Andrew Thompson, Kevin Wilkinson, Joe
Williams, and Faijin Yuan.

[BeamlinePaper.bib]


# Experimental determination of axion signal power of dish antennas and
dielectric haloscopes using the reciprocity approach

J. Egge,111Corresponding author. M. Ekmedžić  A. Gardikiotis,222Now at INFN,
Sezione di Padova, Padova, Italy. E. Garutti  S. Heyminck  C. Kasemann  M.
Kramer  C. Krieger  D. Leppla-Weber  S. Martens  E. Öz  N. Salama  A. Schmidt
H. Wang  G. Wieching

###### Abstract

The reciprocity approach is a powerful method to determine the expected signal
power of axion haloscopes in a model-independent way. Especially for open and
broadband setups like the MADMAX dielectric haloscope the sensitivity to the
axion field is difficult to calibrate since they do not allow discrete
eigenmode analysis and are optically too large to fully simulate. The central
idea of the reciprocity approach is to measure a reflection-induced test field
in the setup instead of trying to simulate the axion-induced field. In this
article, the reciprocity approach is used to determine the expected signal
power of a dish antenna and a minimal dielectric haloscope directly from
measurements. The results match expectations from simulation but also include
important systematic effects that are too difficult to simulate. In
particular, the effect of antenna standing wave and higher order mode
perturbations can be quantified for the first time in a dielectric haloscope.

##  1 Introduction

The axion is an increasingly popular particle candidate for dark matter with
more and more experiments trying to detect it. Axions are a consequence of the
hypothetical Peccei-Quinn mechanism that explains the absence of charge-parity
(CP) violations in the strong interaction, also known as the strong CP problem
[1, 2, 3]. If axions exist, they would not only solve the strong CP problem
but could also be produced in the early universe in sufficient abundance to
constitute all of dark matter. Its mass $m_{a}$ and coupling to standard model
particles is, however, largely unconstrained and experiments have to cover
many orders of magnitude[4]. Experiments that search for axions from the
galactic dark matter halo are called haloscopes and most of them do so via the
coupling of axions to photons which is effectively described by axion-modified
Maxwell’s equations [5]. Together with a strong and static external magnetic
(electric) field ${\bf\it B}_{e}$ (${\bf\it E}_{e}$), the axion field $a$
sources an effective current density ${\bf\it J}_{a}$ that enters Ampère’s law

${\bf\it J}_{a}=\frac{g_{a\gamma}}{Z_{0}}\quantity(\dot{a}{\bf\it
B}_{e}-{\bf\it E}_{e}\crossproduct\gradient{a}),$ (1.1)

where $g_{a\gamma}$ is the axion-photon coupling constant and $Z_{0}$ the
impedance of free space. Throughout this article, SI units are used. Assuming
negligible axion momentum, ${\bf\it J}_{a}$ oscillates with frequency
$\nu_{a}\sim m_{a}$ and in turn sources the axion-induced fields $({\bf\it
E}_{a},{\bf\it H}_{a})$. The traditional haloscope design is a cavity with
resonance frequency $\nu_{a}$ such that the axion-induced fields are amplified
and detected as a power excess above background [6, 7]. In the effort to reach
unexplored parameter space, new approaches are being developed that deviate
from the traditional cavity haloscope. This is because the cavity volume $V$
and resonance frequency $\nu_{a}$ are tightly coupled by $V\sim\nu_{a}^{-3}$
making the cavity impractically large for small axion masses $m_{a}$ or
reducing the signal power, proportional to $V$, too much for large $m_{a}$.
Thus, a lot of new approaches can be thought of as attempting to decouple $V$
from $m_{a}$. One way to do this is open systems like dish antennas [8, 9, 10]
or dielectric haloscopes [11] where the open boundary conditions, to first
order, allow scaling $V$ independently of $\nu_{a}$. A drawback is that these
optically large setups become very hard to simulate. Simulations are, however,
required to predict the axion-induced fields $({\bf\it E}_{a},{\bf\it H}_{a})$
which cannot be measured in the absence of an axion discovery. To not entirely
rely on simulations, one can study the response of the setup to a different
source current density ${\bf\it J}_{R}$, for example from a signal generator
in a reflection measurement. This then excites the reflection-induced fields
$({\bf\it E}_{R},{\bf\it H}_{R})$. Closed resonators usually allow for only a
single, discrete mode at a given frequency and, up to normalization, $({\bf\it
E}_{a},{\bf\it H}_{a})$ and $({\bf\it E}_{R},{\bf\it H}_{R})$ will be the
same. In contrast, open systems allow infinitely many continuous modes at the
same frequency and the axion-induced and reflection-induced fields are not the
same. Here, the reciprocity approach enters where the axion signal power
$P_{\mathrm{sig}}$, instead of depending on the axion-induced fields, is
rewritten to depend on the reflection-induced fields that are excited with
input power $P_{\mathrm{in}}$ [12],

$P_{\mathrm{sig}}=\frac{g_{a\gamma}^{2}}{16Z_{0}^{2}P_{\mathrm{in}}}\absolutevalue{\int_{V_{a}}\differential{V}{\bf\it
E}_{R}\dotproduct\left(\dot{a}{\bf\it B}_{e}-{\bf\it
E}_{e}\crossproduct\gradient{a}\right)}^{2}.$ (1.2)

The axion signal power $P_{\mathrm{sig}}$ now only depends on measurable
quantities directly controlled by the experiment and the axion parameters one
wishes to constrain. The external fields are confined to the conversion volume
$V_{a}$ in the sense that the contribution to $P_{\mathrm{sig}}$ from outside
$V_{a}$ becomes negligible. Reciprocity is a property of linear systems so in
addition to the usual requirements of linear and time-independent material
properties an important assumption is that the axion-modified Maxwell’s
equations are linearized for strong external fields and small axion-photon
coupling [13]. Another assumption is that the reflection-induced fields are
excited via the same port that is used to measure the axion-induced fields.
This port must support only a single mode whereas the rest of the optical
system can be completely arbitrary.

While the reciprocity approach was validated numerically before [14, 12], it
has not been used experimentally. In this article, this will be achieved on a
MADMAX dielectric haloscope test setup, measuring the expected axion signal
power of an open haloscope for the first time. A key ingredient for this is
non-resonant perturbation theory, more commonly known as the non-resonant bead
pull method, which is used to measure the reflection-induced fields [15]. The
article is structured as follows: In section 2 the working principles of a
dielectric haloscope in general and the experimental setup specifically are
described. Section 3 introduces non-resonant perturbation theory and its
application on measuring ${\bf\it E}_{R}$. The simulations used in this study
are briefly described in section 4. The results are presented in section 5.
This is done by successively building up the dielectric haloscope, validating
the measurement procedure on the simplest setup first before finally measuring
$P_{\mathrm{sig}}$ on a minimal dielectric haloscope. Conclusions are then
drawn in section 6.

##  2 Minimal dielectric haloscope

A dielectric haloscope like MADMAX uses many dielectric disks to boost the
axion signal to detectable levels [11]. From the perspective of the axion-
induced fields, each interface emits a traveling wave along its normal
direction. These emissions can then interfere constructively with each other
as well as resonate between the disks, boosting the total signal power. A
plane mirror at one end of the dielectric stack ensures that all emissions
travel toward the receiver system. The stack of dielectric disks plus a plane
mirror is called a booster. The power boost factor $\beta^{2}$ is the
amplification compared to the power $P_{0}$ emitted from just a perfect plane
mirror with area $A$,

$\displaystyle\beta^{2}\equiv\frac{P_{\mathrm{sig}}}{P_{0}}$ (2.1)
$\displaystyle
P_{0}=\frac{g_{a\gamma}^{2}c^{2}\absolutevalue{a_{0}}^{2}\absolutevalue{{\bf\it
B}_{e}}^{2}A}{2Z_{0}}.$ (2.2)

The booster of a minimal dielectric haloscope, shown in figure 1, consists of
one dielectric disk and a plane mirror. A focusing mirror couples the axion-
induced fields to an antenna connected to the receiver chain that then
measures $P_{\mathrm{sig}}$.

Figure 1: Minimal dielectric haloscope setup consisting of an antenna,
focusing mirror, and booster. The antenna is connected to a VNA that measures
the reflection coefficient $\Gamma$. A dielectric bead can be moved in between
or in front of the booster in all three dimensions to map the electric field.

In the reciprocity approach, the vector network analyzer333Rhode & Schwarz
ZVA67 (VNA) represents ${\bf\it J}_{R}$ and sources the reflection-induced
field ${\bf\it E}_{R}$. From here on out, the subscript differentiating
between reflection-induced and axion-induced fields is dropped and ${\bf\it
E}$ refers to the reflection-induced fields unless stated otherwise. The
antenna emits a beam that is focused onto the booster by the focusing mirror.
When the beam enters the booster, it can resonate between the semi-transparent
dielectric disk and plane mirror when their separation is roughly half a
wavelength. This increases ${\bf\it E}$ relative to $P_{\mathrm{in}}$ which
then increases $P_{\mathrm{sig}}$ according to equation (1.2). Inside the
disk, ${\bf\it E}$ has a negative lobe which is suppressed by the disk’s
dielectric constant $\epsilon$ which further increases $P_{\mathrm{sig}}$.
This describes the basic working principle of a dielectric haloscope in the
longitudinal direction. For a complete description and ultimately full signal
power, the behavior in the transverse direction is important.

The coordinate system $(x,y,z)$ is defined as follows: The origin is the
reflection point of the focusing mirror. The optical axis points to the center
of the booster and defines the z-axis. The y-axis points up from the optical
table and the x-axis points towards the wall in figure 1. The external
magnetic field ${\bf\it B}_{e}$ is imagined to point along the y-direction. It
is assumed to be homogeneous over the extent of the booster and decreasing to
negligible magnitude outside the booster. The focusing mirror and antenna
would not be magnetized. The circular horn antenna produces a near-perfect
Gaussian beam, polarized in y-direction, that is redirected and focused by the
off-axis ellipsoidal focusing mirror. The result, ideally, is again a Gaussian
beam converging onto the booster. Gaussian beams are the fundamental mode of
the Hermite-Gaussian modes

$\displaystyle E_{ml}(x,y,z^{\prime})$
$\displaystyle=E_{0}\frac{w_{0}}{w(z)}H_{l}\left(\frac{\sqrt{2}x}{w(z)}\right)H_{m}\left(\frac{\sqrt{2}y}{w(z)}\right)\exp\left({-{\frac{r^{2}}{w(z)^{2}}}}\right)$
(2.3)
$\displaystyle\times\exp\left(-i\left(kz^{\prime}+k{\frac{r^{2}}{2R(z^{\prime})}}-\psi_{ml}(z^{\prime})\right)\right).$

where $r=\sqrt{x^{2}+y^{2}}$ is the radial distance from the optical axis,
$z^{\prime}=z-z_{0}$ the axial distance from the beam waist at $z_{0}$ and of
radius $w_{0}$, $k$ the in-media wave number, and $E_{0}$ the complex
amplitude at $z^{\prime}=0$. The first line in equation (2.3) describes the
transverse beam shape while the second line describes the phase evolution. The
Hermite polynomials $H_{m,l}$ equal unity for $m,l=0$, recovering the Gaussian
beam. The waist radius depends on $z^{\prime}$ as

$w(z^{\prime})=w_{0}\sqrt{1+\left(\frac{z^{\prime}}{z_{R}}\right)^{2}},$ (2.4)

with Rayleigh range $z_{R}=\frac{kw_{0}^{2}}{2}$. The radius of curvature of
the beam’s wavefronts $R(z^{\prime})$ is defined as

$R(z^{\prime})=z^{\prime}\left[1+\left(\frac{z_{R}}{z^{\prime}}\right)^{2}\right].$
(2.5)

The Gouy phase $\psi_{ml}(z^{\prime})$ is

$\psi_{ml}(z^{\prime})=(m+l+1)\arctan{\left(\frac{z^{\prime}}{z_{R}}\right)}.$
(2.6)

The waist radius on the booster side is $\sim$45\text{\,}\mathrm{mm}$$ at
$20\text{\,}\mathrm{GHz}$ resulting in a Rayleigh range of
$z_{R}\sim$45\text{\,}\mathrm{cm}$$. For $z^{\prime}<z_{R}$,
$\psi_{00}(z^{\prime})$ is small, $R(z^{\prime})$ is large, and the phase term
of $E_{ml}(x,y,z^{\prime})$ is well approximated by a plane wave.

The booster reflects the antenna beam from more than one interface which even
under ideal circumstances are not matched to its curvature $R(z)$.
Consequently, the reflected beam is different from the emitted antenna beam.
In reality, geometrical inaccuracies like tilts and surface deformations
distort the beam further. The distortions can be described as exciting higher
order modes (HoM) $E_{m>0,l>0}$. The reflected beam thus contains a fraction
of HoMs. Since the HoMs are orthogonal to the near-perfect fundamental mode
the antenna emits, they are reflected from the antenna instead of being
received. For the HoM, the antenna acts like a mirror and the antenna-booster
system is essentially a leaky cavity. The distance between the antenna and the
booster is $\mathcal{O}(100\lambda)$ and so the harmonics of the antenna-
booster resonance will be very close in frequency. The effect is seen in the
reflection coefficient $\Gamma$ that is measured by the VNA. Its absolute
value is plotted in figure 2(a). The spectrum is dominated by the harmonics of
the antenna-booster resonance while the actual booster resonance is hidden
underneath.

(a)

(b)

Figure 2: Reflection coefficient $\Gamma$ of the minimal dielectric haloscope.
(a): Magnitude of $\Gamma$ over frequency. The spectrum is dominated by the
antenna-booster resonances which are removed via time gating (dashed line).
(b): Fourier transform of (a) a gate (dashed) is applied to filter out
antenna-booster resonances.

A common method to filter out undesired reflections from a measurement is time
gating. By Fourier transforming $\Gamma$ into the time domain, figure 2(b),
the antenna-booster resonance manifests itself as successive peaks separated
by an interval corresponding to twice the distance between antenna and
booster. Applying a gate on the main peak and transforming back into the
frequency domain isolates the signal that is not reflected by the antenna,
removing the antenna-booster resonances. Time gating relies, however, on the
phase information of the signal. Since the axion signal is measured only in
power, it cannot be time-gated and antenna reflections cannot simply be
ignored. The reciprocity approach offers a model-independent way to measure
the expected signal power, regardless of undesired reflections or not. To do
that, one needs to measure the reflection-induced electric field ${\bf\it E}$
throughout the booster.

##  3 Bead pull method

Non-resonant perturbation theory [15], also known as bead pull method, is a
natural choice for measuring the reflection-induced electric field ${\bf\it
E}$. It relates the change in reflection coefficient $\Gamma$ from introducing
an object, in this case a dielectric bead, into an arbitrary optical system to
the electric field at that object’s position. In its most general form, one
finds

$\Delta\Gamma=-\frac{i\epsilon_{0}\omega(\epsilon_{B}-1)}{4P_{\mathrm{in}}}\int_{V_{B}}\differential{V}{\bf\it
E}_{B}\cdot{\bf\it E},$ (3.1)

where ${\bf\it E}_{B}$ is the electric field in the presence of the bead with
relative permittivity $\epsilon_{B}$ and the integration is performed over the
volume of the bead $V_{B}$. A background medium of $\epsilon=1$ is already
assumed here. While (3.1) is an exact relation applicable to an arbitrary
object of any size, it is not very usable since it includes the electric field
inside the bead ${\bf\it E}_{B}$. To relate the internal field ${\bf\it
E}_{B}$ to the exciting field ${\bf\it E}$ is a common problem in scattering
theory. For a homogeneous dielectric sphere with a radius $r_{B}$ this can be
solved analytically by Mie theory [16]. For any radius but $r_{B}\ll\lambda$
the expressions quickly become unwieldy. In particular, the fact that ${\bf\it
E}_{B}$ is not homogeneous over the extent of the bead complicates matters
significantly as the integrand of equation (3.1) cannot be separated. A
practical approach is then to define the bead factor

$\delta_{e}=\frac{V_{B}(\epsilon_{B}-1)}{\alpha_{0}}\frac{\langle{\bf\it
E}_{B}\cdot{\bf\it E}\rangle}{{\bf\it E}^{2}},$ (3.2)

where the electric field in the denominator is evaluated at the bead’s center,
the angled brackets denote the average over $V_{B}$, and $\alpha_{0}$ is the
polarizability of a dielectric sphere in a static, homogeneous electric field

$\alpha_{0}=4\pi r_{B}^{3}\frac{\epsilon_{B}-1}{\epsilon_{B}+2}.$ (3.3)

Another quantity similar to $\delta_{e}$ is

$\delta_{c}=\frac{V_{B}(\epsilon_{B}-1)}{\alpha_{0}}\frac{\langle{\bf\it
E}_{B}\cdot{\bf\it E}^{*}\rangle}{\absolutevalue{{\bf\it E}}^{2}},$ (3.4)

which will be needed in the dish antenna and dielectric haloscope setup to
better deconvolve $\delta_{e}$ from the measured $\Delta\Gamma$. Generally,
$\delta_{e}$ and $\delta_{c}$ are functions of both position and frequency. In
the limit of an infinitely small bead, $\delta_{e}$ and $\delta_{c}$ become
unity

$\lim_{kr_{B}\to 0}\delta_{e,c}=1.$ (3.5)

Mie theory is used to compute $\delta_{e,c}$ for finite radii. The analytical
expressions are cumbersome and can be found in the appendix B. Equation (3.1)
can then be rewritten as

$\Delta\Gamma=\frac{\epsilon_{0}\delta_{e}\alpha_{0}\omega}{4P_{\mathrm{in}}}{\bf\it
E}^{2}.$ (3.6)

It is worth noting that ${\bf\it E}^{2}$, just as $\Delta\Gamma$, is a complex
quantity with phase information. It also contains contributions from all
spatial components whereas only the component parallel to ${\bf\it B}_{e}$
will contribute to $P_{\mathrm{sig}}$. Thus an important assumption is that
${\bf\it E}$ is polarized along the would-be external magnetic field ${\bf\it
B}_{e}$. Since the cross-polarization of the antenna is
$<$30\text{\,}\mathrm{dB}$$, and the alignment of optical components is well
controlled, the contribution of cross-polarization to $\Delta\Gamma$ is
expected to be negligible. Finally, and quite conveniently, $\Delta\Gamma$
inversely depends on the input power $P_{\mathrm{in}}$ so that it cancels when
inserting equation (3.6) into (1.2).

The bead used in this study is made of alumina. Its relative permittivity was
measured to be $\epsilon_{B}=$9.23\pm 0.05$$ by inserting it into a resonant
cavity with known dimensions and observing shifts in resonance frequencies.
This suggests some impurities as literature puts alumina closer to
$\epsilon\sim 10$. Its radius of $r_{B}=$1.46\pm 0.01\text{\,}\mathrm{mm}$$ is
much smaller than the beam waist $w_{0}$ and roughly $\lambda/10$.
Consequently, ${\bf\it E}$, but not ${\bf\it E}_{B}$, is essentially
homogeneous in transverse direction over the extent of the bead while in
longitudinal direction small but non-negligible variation is expected. The
bead is attached to a string made from high-modulus polyethylene (HMPE) with a
diameter of $0.1\text{\,}\mathrm{mm}$. Judging from the relatively low
$\epsilon$ of HMPEs and the tiny volume of the string, a negligible effect of
the string on $\Delta\Gamma$ is expected. The bead can be positioned in three
dimensions. In x-direction, the string is pulled against a counterweight by a
stepper motor with an estimated accuracy of
$<$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. Two linear translation
stages444Thorlabs LTS300 move the whole bead pull setup in y and z-direction
with an accuracy of $<$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$.

##  4 Simulation

The setups under consideration are simulated using COMSOL Multiphysics®, a
finite element simulation software. As the setups are optically large, 2D
axisymmetry is used to make simulation possible in the first place. This means
that only azimuthally symmetric geometry can be simulated. The focusing mirror
breaks this symmetry and thus the simulation does not include the focusing
mirror and antenna. Instead, the setups are simulated using the scattered
field formulation where the background field is set to the Gaussian beam that
is expected to be produced by the antenna and focusing mirror. While saving
considerable computational cost, the simulation cannot include the antenna-
booster resonances. Nevertheless, one can still compare measurement and
simulation by filtering out the antenna-booster resonances from the
measurement using time gating. To simulate the linearly polarized fields
produced by the antenna in 2D axisymmetric mode, all relevant field components
$X$ are decomposed into a left and right circularly polarized part [17]:

$X(r,\phi,z)=X^{+}(r,z)e^{-i\phi}+X^{-}(r,z)e^{+i\phi}$ (4.1)

Only the azimuthally symmetric components $X^{\pm}(r,z)$ are simulated and
later added according to (4.1). The entire domain is surrounded by perfectly
matched layers to absorb any outgoing radiation. The dish antenna setup is
simulated by adding a plane mirror using an impedance boundary condition. The
axion signal power is calculated using (1.2). For the dielectric haloscope, a
disk is added with the appropriate dimensions.

##  5 Results

The reciprocity approach is tested on three different setups. The first setup
consists of just the antenna and focusing mirror and characterizes the antenna
beam as well as bead pull systematics. Next, a plane mirror is added. This
configuration is also known as a dish antenna. Finally, a dielectric disk
completes the minimal dielectric haloscope. The input power $P_{\mathrm{in}}$
is arbitrarily set to $1\text{\,}\mathrm{W}$ as its true value, which is not
known precisely, cancels in the end. As ${\bf\it E}$ and ${\bf\it B_{e}}$ are
assumed to be polarized in the y-direction, it is sufficient to use the scalar
fields $E={\bf\it E}\dotproduct{\bf\it y}$ and $B_{e}={\bf\it
B}_{e}\dotproduct{\bf\it y}$.

###  5.1 Antenna beam

Figure 3: Electric field $E$ produced by the antenna and focusing mirror. Top:
$|\Gamma|^{2}$ over frequency. The markers indicate the frequencies at which
$E$ is shown below from left to right. Middle row: transverse beam profile at
three different frequencies. Bottom row: Longitudinal beam profile. The dashed
lines indicate the beam waist position and size.

Before measuring $P_{\mathrm{sig}}$ on the minimal dielectric haloscope, the
bead pull method is tested on a simpler setup consisting of just the antenna
and focusing mirror. The relevant antenna and focusing mirror characteristics
are known from independent measurements and ${E}$ can be benchmarked against
expectations. Additionally, the measured beam parameters will serve as an
input to simulations. RF absorbers are put in place of the booster to emulate
free space. The reflection coefficient $\Gamma$ for this setup can be seen in
the top panel of figure 3. The non-zero baseline and long oscillations come
from reflections at the antenna input. The short oscillations come from
reflections at the RF absorbers. These reflections are irrelevant as the
absorbers will later be replaced by the booster. Thus $\Gamma$ is time-gated
to exclude absorber reflections. The bead is then moved across the beam at
different heights and distances from the focusing mirror. Measurements of
$\Gamma$ where the bead is well outside the beam are used as a reference for
$\Delta\Gamma$ from which ${E}$ is calculated using (3.6). The detailed
procedure that accounts for signal drift, as well as potential branching
issues when taking the complex square root of $\Delta\Gamma$, is described in
the appendix A. A bead factor of $\delta_{e}=1$ is assumed for now. Deviations
from this assumption are discussed later.

Figure 3 shows the measured beam in the $xy$ and $xz$ plane at three different
frequencies as indicated by the markers in the spectrum above. The dashed line
shows the beam waist $w(z)$ that was extracted by fitting a Gaussian beam
${E}_{00}$ separately for each frequency to the measured field. The overlap
with the fundamental mode is found to be $>$96\text{\,}\mathrm{\char
37\relax}$$.

(a)

(b)

Figure 4: (a): Beam waist size $w_{0}$ (top panel) and position $z_{0}$
(bottom panel) over frequency as measured by the bead pull method (solid
line). The dashed line shows the expectation from the antenna and focusing
mirror characteristics. (b): Bead factor $\delta_{e}$ from measurement (solid
line) and FEM simulation (dashed line).

The fitted beam waist size $w_{0}$ and position $z_{0}$ are shown in figure
4(a) together with the expectation from the antenna radiation pattern that was
measured independently in a reverberation chamber by the Max Planck Institute
for Radio Astronomy who also designed the antenna. The radiation pattern gives
the frequency-dependent beam waist size $w_{A}$ and position $z_{A}$ on the
antenna side which together with the focal length $f$ of the focusing mirror
can be used to calculate $w_{0}$ and $z_{0}$ via the thin lens equation,

$\frac{w_{A}^{2}}{w_{0}^{2}}=\frac{z_{A}-f}{z_{0}-f}.$ (5.1)

The antenna radiation pattern gives $z_{A}$ relative to the antenna’s aperture
whereas for equation (5.1) the absolute distance to the reflection point of
the focusing mirror is needed which is not known precisely. To obtain the good
match of figure 4(a), a frequency-independent offset of
$\sim$6\text{\,}\mathrm{mm}$$ is added to $z_{A}$, still within the estimated
uncertainty on the absolute positions of antenna, focusing mirror and bead.
The focal length also needs to be scaled by $4.8\text{\,}\mathrm{\char
37\relax}$ from the geometric value of $f=$352.5\text{\,}\mathrm{mm}$$. This
is a bit surprising since the focusing mirror was machined with
$\sim$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ precision. It is unclear
where this mismatch could come from.

So far $\delta_{e}=1$ was assumed. The axion signal power $P_{\mathrm{sig}}$,
however, depends on the true value of $E$ and thus $\delta_{e}$. Over the
extend of the bead, ${E}$ can be assumed to be a plane wave for which
$\delta_{e}$ is independent of bead position. Consequently, the shape of $E$
should not be affected by $\delta_{e}$. In this simple setup without a
booster, the power contained in the beam $P_{\mathrm{beam}}\sim
P_{\mathrm{in}}$ since inefficiencies coming from the antenna and focusing
mirror are negligible. For the antenna, this is apparent from figure 3 where
$|\Gamma|^{2}<$-18\text{\,}\mathrm{dB}$$. Focusing mirror inefficiencies come
from the conduction loss of aluminium and losses due to surface roughness
which can be estimated by Ruze’s equation [18]. Using a conductivity of
$\sigma_{Al}=$5\text{\times}{10}^{7}\text{\,}\mathrm{S}\text{\,}{\mathrm{m}}^{-1}$$
and surface errors of $\sigma_{RMS}\sim$10\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ yields losses of less than $-30\text{\,}\mathrm{dB}$ and
$-40\text{\,}\mathrm{dB}$, respectively. $P_{\mathrm{beam}}$ can be calculated
by integrating the measured ${E}$ in the $xy$ plane:

$P_{\mathrm{beam}}=\frac{1}{2Z_{0}}\int\differential{A}\left\lvert{E}\right\rvert^{2}.$
(5.2)

Setting $P_{\mathrm{beam}}=P_{\mathrm{in}}$, one then finds from equation
(3.6)

$\delta_{e}^{+}=\frac{2}{Z_{0}\alpha_{0}\omega}\int\differential{A}\left\lvert\Delta\Gamma\right\rvert,$
(5.3)

where the + in superscript indicates the assumption of a plane wave traveling
in the positive z-direction. Figure 4(b) shows $\delta_{e}^{+}$ obtained from
measurement via (5.3) and analytically from Mie theory, see equation (B.1) in
the appendix. The shaded ribbon is the uncertainty on the measured
$\delta_{e}^{+}$ that is in part estimated from observed fluctuations in the
measured $\Delta\Gamma$ but largely due to uncertainty in $\alpha_{0}$ which
is in turn dominated by uncertainty in the bead radius $r_{B}$. The agreement
with the result from Mie theory is better than $\sim$8\text{\,}\mathrm{\char
37\relax}$$. The uncertainty from Mie theory again stems largely from the
uncertainty in the bead’s radius. For now, the remaining mismatch between
measurement and simulation will be treated as a systematic uncertainty on
$\delta_{e}$ that later directly translates to an uncertainty on
$P_{\mathrm{sig}}$.

###  5.2 Dish antenna

Figure 5: Electric field in front of the dish antenna with a diameter of
$300\text{\,}\mathrm{mm}$ positioned at $z\sim$787\text{\,}\mathrm{mm}$$. Top
panel: Reflection coefficient of the setup. The markers indicate the
frequencies at which the electric field is shown below from left to right.
Middle row: Transverse field profile at $z=$783\text{\,}\mathrm{mm}$$ with
increasing amount of distortion due to antenna-booster resonances. Bottom row:
Corresponding longitudinal field profile at $y=$0\text{\,}\mathrm{mm}$$ with
visible standing wave pattern. The plane mirror is shown as a dark gray band.

Now that the bead pull method has been benchmarked with a known and relatively
simple electric field, the next step is to add a plane mirror forming a dish
antenna haloscope. Without any magnetized interface, there is no axion-photon
conversion as integrating a traveling wave per equation (1.2) evaluates to
zero for a homogeneous ${B}_{e}$. By adding the plane mirror, ${E}$ becomes a
standing wave for which the integral in (1.2) is non-zero. The plane mirror is
put at $z_{m}=$786.5\pm 0.1\text{\,}\mathrm{mm}$$ perpendicular to the beam
propagation and has a diameter of $300\text{\,}\mathrm{mm}$. Its position is
estimated from the linear motor stage position by moving the bead as close as
possible to the mirror. As discussed before, distortions of the beam from its
original Gaussian shape lead to antenna-booster resonances. Figure 5 shows the
magnitude of ${E}$ in the $xy$ and $xz$ plane as measured in front of the
plane mirror at three different frequencies. Off antenna-booster resonance,
where $\absolutevalue{\Gamma}$ is maximal, the field appears relatively
undisturbed from the Gaussian beam while on resonance, low
$\absolutevalue{\Gamma}$, the contribution from HoMs is clearly visible.
Having measured ${E}$, one can now expect two opposing effects that the
antenna-booster resonances have on $P_{\mathrm{sig}}$. First, the overall
magnitude of ${E}$ increases on resonance which can boost the signal power.
Second, the higher order modes generally have a poor overlap with a uniform
$a{B}_{e}$ which would decrease $P_{\mathrm{sig}}$. The z-dependence of ${E}$
can be approximated by plane wave propagation since the measurement range is
smaller than the Rayleigh range of $\sim$45\text{\,}\mathrm{cm}$$. By
integrating ${E}$ in the transverse direction, the analysis can be reduced to
one dimension. The one-dimensional electric field amplitude is defined as

$E_{1D}=\int\differential{A}{E}.$ (5.4)

Note that $E_{1D}$ has dimensions of $\mathrm{V}\text{\,}\mathrm{m}$.
Crucially, the complex electric field $E$ is integrated in (5.4). With the
plane mirror at one end, $E_{1D}$ is a superposition of two plane waves
traveling in opposite directions,

$E_{1D}=E_{+}e^{-ikz}+E_{-}e^{+ikz}.$ (5.5)

Figure 6(a) shows the measured $E_{1D}$ over z at a selected frequency. The
errors are estimated from fluctuations of $\Delta\Gamma$ at the edges of the
booster. The measurement is clearly similar to a standing wave with the
exception that at the nodes a non-zero offset remains. The finite size of the
bead effectively averages over the extent of the bead, similar to a moving
average. This is accounted for by the bead factor $\delta_{e}$. To obtain the
true magnitude of $E_{1D}$, the measured $E_{1D}$ has to be deconvolved from
$\delta_{e}$ which is now a function of both frequency and position. The
quantity that is actually measured is $\Delta\Gamma$. Using (5.5) and the fact
that for a single plane wave $\delta_{e}^{+}$ is independent of position and
direction, $\Delta\Gamma$ can be written as

$\frac{4P_{\mathrm{in}}\Delta\Gamma}{i\omega\alpha_{0}}=\delta_{e}^{+}\left(E_{+}^{2}e^{-2ikz}+E_{-}^{2}e^{+2ikz}\right)+2\delta_{c}^{+}E_{+}E_{-},$
(5.6)

with the additional bead factor $\delta_{c}^{+}$ from (3.4) where again the +
in superscript indicates the assumption of a single plane wave for which
$\delta_{c}^{+}$ is independent of position and direction. It is computed
analytically from Mie theory, see equation (B.2). By fitting equation (5.6) to
the measured $\Delta\Gamma$, the true $E_{1D}$ can be extracted. A transfer
matrix formalism [19] is used to relate the forward and backward amplitudes
$E_{+}$ and $E_{-}$ with each other. In this case, by fixing the conductivity
of the mirror to $\sigma_{Al}$, only the mirror position $z_{m}$ and forward
amplitude $E_{+}$ are left as free parameters for the fit. Fitting is
performed separately for each frequency using nonlinear least-squares. This
means that the fit does not need to include complicated frequency dependencies
as these will be absorbed into the fitted $E_{+}$. The fitted mirror position
averaged over frequency is found to be $z_{m}=$786.63\pm
0.01\text{\,}\mathrm{mm}$$ showing both the accuracy of the bead pull method
as well as the validity of plane wave propagation since deviations from plane
wave propagation would likely manifest themselves as a fictitious frequency-
dependence of physical lengths. The uncertainty on $E_{+}$ is given by the
covariance matrix of the fit and is less than $8\text{\,}\mathrm{\char
37\relax}$.

(a)

(b)

Figure 6: (a): Transversely integrated electric field over $z$ at
$\nu\sim$19.44\text{\,}\mathrm{GHz}$$. The measurements are fitted by equation
(5.6) (dashed line) allowing to deconvolve finite size effects of the bead
from the measurement (dashed-dotted line). (b) Boost factor $\beta^{2}$ of the
dish antenna from measurement and simulation.

The final step in calculating $P_{\mathrm{sig}}$ is to compute

$P_{\mathrm{sig}}=\frac{g_{a\gamma}^{2}\omega^{2}}{16P_{\mathrm{in}}}\absolutevalue{\int_{-\infty}^{z_{m}}\differential{z}E_{1D}aB_{e}}^{2}.$
(5.7)

The external magnetic field and axion field $aB_{e}$ are set to one over the
extent of the booster and decrease as $\sim\cos^{2}(\frac{z}{6\lambda})$ to
zero outside of the booster. This makes sure that contributions to
$P_{\mathrm{sig}}$ from magnetic field inhomogeneities remain negligible.
Equation (5.7) is evaluated numerically. The resulting axion signal power of
the dish antenna is shown in figure 6(b), parameterized as the boost factor
$\beta^{2}$ to remove the dependency on $B_{e}$ and axion parameters. The
uncertainty in $\beta^{2}$ is shown as a shaded band. The uncertainties of the
input parameters to $E_{1D}$, namely $E_{+}$, $z_{m}$ and $\sigma_{Al}$, are
propagated to $P_{\mathrm{sig}}$ using a Monte Carlo method. The uncertainty
in $\alpha_{0}$ and $\delta_{e}$ directly translate to $P_{\mathrm{sig}}$
bringing the final uncertainty to less than $15\text{\,}\mathrm{\char
37\relax}$. Shown as well is the boost factor obtained from numerical
simulation that does not include the antenna-booster resonances as well as a
measurement where the antenna-booster resonances are filtered out using time
gating. The baseline of measurement and numerical simulation both match,
indicating a good calibration of the bead-pull method. Note that the reduction
of the boost factor from an ideal value of one to $\sim 0.3$ is purely from
the poor power coupling of the given Gaussian beam to the uniform $a{B}_{e}$
and not from losses. By increasing the waist size, one could obtain a
theoretical maximum power coupling of $\sim 0.8$. Such a beam, however, would
have a significant spill-over at the mirror contributing further to antenna-
booster resonances. Even with the more conservative waist size of this setup,
the antenna-booster resonances cause a maximum reduction of around
$50\text{\,}\mathrm{\char 37\relax}$ from the baseline. While this is still a
$\mathcal{O}(1)$ effect that is hardly visible in a logarithmic exclusion
plot, one has to keep in mind that dish antennas do not have a lot of room for
error and with a bit less care one could, at specific frequencies, end up with
zero signal power. The upshot is that conversely, a decent boost in signal
power is also possible, in this case of up to an order of magnitude. Overall,
it should be noted that a realistic dish antenna has a far more interesting
frequency response than it is usually assumed to have.

###  5.3 Dielectric haloscope

To complete the minimal dielectric haloscope a sapphire disk is added in front
of the mirror separated by mechanical spacers with a distance between disk and
mirror of $d_{v}=$7.95\pm 0.05\text{\,}\mathrm{mm}$$. The disk has radius of
$300\text{\,}\mathrm{mm}$ and a thickness of $d_{\epsilon}=$1.00\pm
0.05\text{\,}\mathrm{mm}$$. The relative permittivity of sapphire in the
radial direction was measured to be $\epsilon=$9.3\pm 0.1$$ at
$20\text{\,}\mathrm{GHz}$ [20]. The field is measured between the disk and
mirror as well as in front of the disk. Figure 7 shows the field distribution
of $|E|$ in the booster. The chosen frequencies are all on the booster
resonance but not all on an antenna-booster resonance. In the left panel, on
booster resonance but off antenna-booster resonance, the field is undisturbed.
The middle and right panels show how the field is increasingly perturbed as
the booster resonance and HoM antenna-booster resonance mix.

Figure 7: Electric field inside and in front of the dielectric haloscope. Top
panel: Reflection coefficient of the setup. The markers indicate the
frequencies at which the electric field is shown below from left to right.
Middle row: Transverse field profile at $z\sim$784\text{\,}\mathrm{mm}$$,
between disk and mirror. Bottom row: Longitudinal field profile between disk
and mirror and in front of the disk at $y=$0\text{\,}\mathrm{mm}$$. The field
is strongest between the disk (light gray band) and the mirror (dark gray
band).

The procedure to calculate $P_{\mathrm{sig}}$ for the dielectric haloscope is
analogous to the dish antenna case. As before, the measured $E$ is integrated
in the transverse direction first to obtain $E_{1D}$ which is again fitted by
equation (5.6). In the case of the minimal dielectric haloscope, the transfer
matrix formalism relates the forward and backward amplitudes $E_{+}$ and
$E_{-}$ not only in front of the plane mirror but also inside the disk and in
front of the disk. For this, the additional fixed parameters are the
permittivity and thickness of the disk. The plane mirror position is also
fixed to the value obtained previously from the dish antenna setup. This
leaves the position of the dielectric disk $z_{\epsilon}$ as well as the
forward amplitude $E_{+}$ between disk and mirror as fit parameters. The
measurement and fit are shown in figure 8(a). The fitted disk position varied
less than $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ over frequency, again
showing remarkable accuracy. As a cross-check, the plane mirror position is
also fitted in which case it agrees with the dish antenna value within
$\sim$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. Thus plane wave propagation
seems to be valid also for the minimal dielectric haloscope. As before, all
the complicated effects over frequency and transverse directions are absorbed
into $E_{+}$. The uncertainty of $E_{+}$ from the fit is less than
$5\text{\,}\mathrm{\char 37\relax}$.

(a)

(b)

Figure 8: (a): Transversely integrated electric field over $z$ at
$\nu\sim$19.7\text{\,}\mathrm{GHz}$$. The measurements are fitted including
finite size effects of the bead (dashed line) and then deconvolved (dashed-
dotted line). (b): Boost factor $\beta^{2}$ of the dielectric haloscope from
measurement and simulation.

With $E_{1D}$ extracted from the fit, $P_{\mathrm{sig}}$ is calculated via
equation (5.7) and shown as the boost factor in figure 8(b). The uncertainty
on $\beta^{2}$ is less than $22\text{\,}\mathrm{\char 37\relax}$, a bit higher
than in the dish antenna case. This is attributed to the fact that more
parameters enter the transfer matrix formalism, each with some uncertainty
that is propagated to $\beta^{2}$ via Monte Carlo method. Shown as well is the
result from the 2D axisymmetric FEM simulation and a time-gated measurement
that, again, do not include the antenna-booster resonances which lead to a
deviation of up to $\sim$50\text{\,}\mathrm{\char 37\relax}$$ between
measurement and simulation. Overall, the effect of the antenna-booster
resonances on $P_{\mathrm{sig}}$ seems tolerable. Extending the dielectric
haloscope to more disks comes with its own challenge of increasingly tighter
tolerances on geometric inaccuracies [21]. In addition, it might not be
practical to measure the electric field everywhere in a many disks setup
especially when considering a cryogenic setup. One might imagine a hybrid
approach where a limited amount of bead pull measurements, for example in
front of the booster, serve as input to a more sophisticated model capable of
simulating the booster. This would still massively reduce computational
complexity as the antenna and focusing mirror need not be included in the
model while keeping complexity in the measurement manageable.

##  6 Conclusion

This study applied a new approach to obtain the first realistic axion signal
power of a dish antenna and dielectric haloscope. With the reciprocity
approach, measurements of the reflection-induced electric field can be used to
directly calculate the expected axion signal power with minimal model input.
The electric field was measured with the bead pull method on three different
setups. The first and simplest setup served as a benchmark test of the bead
pull system and provided important input on the antenna, focusing mirror, and
bead properties used in the other two setups. In the dish antenna and minimal
dielectric haloscope, the baseline of the signal power agrees with the
expectation from simulation within $10\text{\,}\mathrm{\char 37\relax}$. The
antenna-booster resonances, however, add deviations of up to
$\sim$50\text{\,}\mathrm{\char 37\relax}$$ from the baseline. The effect of
the antenna-booster resonances can now, for the first time in any open
haloscope, be fully accounted for. In particular, the reciprocity approach
removes the need for a full simulation of the setup which is often not
possible when considering optically large setups like dish antennas and
dielectric haloscopes. Instead, only a simplified local model is needed to
interpolate and deconvolve the measured electric field such that a more
precise result can be obtained. In the future, this method will be applied on
a larger dielectric haloscope capable of exploring new parameter space for
axions.

## Acknowledgments

The MADMAX collaboration made this work possible by providing the necessary
hardware and valued feedback. Special thanks go to Olivier Rossel for his view
from a unique vantage point with both useful and refreshing insights. This
work is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy, EXC 2121, Quantum Universe
(390833306).

##  Appendix A Signal processing

(a)

(b)

(c)

(d)

Figure 9: Signal processing steps shown here on the dielectric haloscope
setup. (a): Real and imaginary part of $\Gamma$ over bead position $x$. The
baseline reference without the bead $\Gamma_{0}$ is obtained from the
measurements at the edges. (b): Fourier transform of $\Delta\Gamma$. A gate is
applied to filter out noisy components. (c,d): Raw and time-gated
$\Delta\Gamma$ over frequency at the beam’s center (c) or outside the beam
(d).

The VNA measures the complex reflection coefficient $\Gamma_{B}(\nu,x,y,z)$
over frequency $\nu$ for each bead position $x,y,z$ resulting in a four-
dimensional data set. This includes bead positions outside the beam that serve
as the reference measurement without the bead $\Gamma_{0}$. Ideally, one would
only need one reference measurement. However, since the setup is subject to
long-term drifts due to changes in temperature, $\Gamma_{0}$ should be as
close as possible in time to each $\Gamma_{B}$. This is achieved by always
scanning across the full beam in the x-direction for each $y$ and $z$
position. Thus one obtains several reference measurements, left and right of
the beam, for each $y$ and $z$ position. To account for drift during a scan in
the x-direction, a linear fit between the left and right reference
measurements interpolates $\Gamma_{0}$ for all $x$ positions. Figure 9(a)
shows $\Gamma_{B}$ and $\Gamma_{0}$ over $x$. Next, the difference is
obtained,

$\Delta\Gamma=\Gamma_{B}-\Gamma_{0}.$ (A.1)

Still, $\Delta\Gamma$ contains the effects of drift. This can be seen by
Fourier transforming $\Delta\Gamma$ into the time domain shown in figure 9(b).
The main peak corresponds to the bead position. Any feature earlier than that
would ideally subtract out as it is unaffected by the bead. Removing these
earlier features by time gating thus further reduces the effect of drifts and
noise without losing information on ${E}$. Features after the main peak
contain, however, the effect of additional antenna reflections and should
ideally be preserved. Since each peak becomes successively smaller and
noisier, the maximum number of peaks included in the time gate is six. This
helps to further reduce noise without losing too much information on the
antenna-booster resonances. To remove the antenna-booster resonance entirely,
the gate would only include the main peak. Figure 9(c) shows $\Delta\Gamma$ in
the beam’s center before and after time gating. The difference is relatively
small. Outside the beam, however, time gating helps to reduce noise
significantly, as can be seen in figure 9(d). This is important because when
calculating $E$ from $\Delta\Gamma$ via equation (3.6), one needs to take the
square root of the complex $\Delta\Gamma$ and one can introduce phase jumps of
$\pi$, which represents an unwrapping problem. The phase of $E$ is expected to
vary slowly compared to the sampling density in the transverse $xy$ plane and
a 2D unwrapping algorithm based on [22] is applied separately to each $xy$
slice. Without any time gating, the algorithm often fails as noise makes
unwrapping, particularly in two dimensions, notoriously difficult. As the
axion signal power depends on the complex $E$ including phase, a failed unwrap
can have a significant effect on $P_{\mathrm{sig}}$. Figure 10 shows the
unwrapped real and imaginary part of $E$ at a frequency with significant HoM
contribution.

Figure 10: Real and imaginary part of the electric field between disk and
mirror at $19.88\text{\,}\mathrm{GHz}$.

##  Appendix B Bead factor from Mie theory

Mie theory analytically treats the scattering of a plane wave off spherical
objects. First, the exciting electric field ${\bf\it E}$ and internal electric
field ${\bf\it E}_{B}$ are expanded in terms of vector spherical harmonics
following standard procedure [16]. Using the orthogonality of vector spherical
harmonics, the bead factors $\delta_{e}^{+}$ and $\delta_{c}^{+}$ can be
expressed as

$\delta_{e}^{+}=\sum_{n=1}^{\infty}(-1)^{n}\left(K_{n}c_{n}-L_{n}d_{n}\right),$
(B.1) $\delta_{c}^{+}=\sum_{n=1}^{\infty}\left(K_{n}c_{n}+L_{n}d_{n}\right).$
(B.2)

The coefficients $c_{n}$, $d_{n}$ are

$c_{n}=\frac{j_{n}(x)[xh_{n}^{(1)}(x)]^{\prime}-h_{n}^{(1)}(x)[xj_{n}(x)]^{\prime}}{j_{n}(mx)[xh_{n}^{(1)}(x)]^{\prime}-h_{n}^{(1)}(x)[mxj_{n}(mx)]^{\prime}}$
(B.3)

and

$d_{n}=\frac{mj_{n}(x)[xh_{n}^{(1)}(x)]^{\prime}-mh_{n}^{(1)}(x)[xj_{n}(x)]^{\prime}}{m^{2}j_{n}(mx)[xh_{n}^{(1)}(x)]^{\prime}-h_{n}^{(1)}(x)[mxj_{n}(mx)]^{\prime}},$
(B.4)

with the spherical Bessel of the first kind $j_{n}$, first spherical Hankel
function $h_{n}^{(1)}$, size parameter $x=kr_{B}$, and relative refractive
index $m=\sqrt{\epsilon_{B}}$. The prime denotes differentiation with respect
to the argument in parentheses. The coefficients $K_{n}$ and $L_{n}$ are

$K_{n}=\frac{2n+1}{m^{2}-1}x^{2}\left[j_{n-1}(x)j_{n}(mx)-mj_{n}(x)j_{n-1}(mx)\right]$
(B.5)

and

$\displaystyle L_{n}$
$\displaystyle=\frac{x^{2}}{m^{2}-1}\\{(n+1)[j_{n-2}(x)j_{n-1}(mx)-mj_{n-1}(x)j_{n-2}(mx)]$
(B.6) $\displaystyle+n[j_{n}(x)j_{n+1}(mx)-mj_{n+1}(x)j_{n}(mx)]\\}.$

The bead factors are then computed numerically in Julia using the
SpecialFunctions.jl package555https://specialfunctions.juliamath.org/stable/
and have been checked against COMSOL FEM simulations.

## References

  * [1] R.D. Peccei and H.R. Quinn, _$\mathrm{CP}$ conservation in the presence of pseudoparticles_, _Phys. Rev. Lett._ 38 (1977) 1440.
  * [2] F. Wilczek, _Problem of strong $p$ and $t$ invariance in the presence of instantons_, _Phys. Rev. Lett._ 40 (1978) 279.
  * [3] S. Weinberg, _A new light boson?_ , _Phys. Rev. Lett._ 40 (1978) 223.
  * [4] Y.K. Semertzidis and S. Youn, _Axion dark matter: How to see it?_ , _Science Advances_ 8 (2022) eabm9928 [https://www.science.org/doi/pdf/10.1126/sciadv.abm9928].
  * [5] P. Sikivie, _Experimental tests of the ”invisible” axion_ , _Phys. Rev. Lett._ 51 (1983) 1415.
  * [6] ADMX Collaboration, _Axion dark matter experiment: Run 1b analysis details_ , _Physical Review D_ 103 (2021) .
  * [7] J. Choi, S. Ahn, B. Ko, S. Lee and Y. Semertzidis, _Capp-8tb: Axion dark matter search experiment around $6.7\text{ }\text{ }\mu\mathrm{eV}$_, _Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment_ 1013 (2021) 165667.
  * [8] D. Horns, J. Jaeckel, A. Lindner, A. Lobanov, J. Redondo and A. Ringwald, _Searching for WISPy cold dark matter with a dish antenna_ , _Journal of Cosmology and Astroparticle Physics_ 2013 (2013) 016.
  * [9] F. Bajjali, S. Dornbusch, M. Ekmedžić, D. Horns, C. Kasemann, A. Lobanov et al., _First results from brass-p broadband searches for hidden photon dark matter_ , _Journal of Cosmology and Astroparticle Physics_ 2023 (2023) 077.
  * [10] DOSUE-RR Collaboration collaboration, _Search for dark photon dark matter in the mass range $74--110\text{ }\text{ }\mathrm{\mu}\mathrm{eV}$ with a cryogenic millimeter-wave receiver_, _Phys. Rev. Lett._ 130 (2023) 071805.
  * [11] A. Caldwell, G. Dvali, B. Majorovits, A. Millar, G. Raffelt, J. Redondo et al., _Dielectric haloscopes: A new way to detect axion dark matter_ , _Physical Review Letters_ 118 (2017) .
  * [12] J. Egge, _Axion haloscope signal power from reciprocity_ , _Journal of Cosmology and Astroparticle Physics_ 2023 (2023) 064.
  * [13] J.A. Dror, H. Murayama and N.L. Rodd, _Cosmic axion background_ , _Phys. Rev. D_ 103 (2021) 115004.
  * [14] K.M. Schulz, D. Jalas, A.Y. Petrov and M. Eich, _Reciprocity approach for calculating the purcell effect for emission into an open optical system_ , _Opt. Express_ 26 (2018) 19247.
  * [15] C. Steele, _A nonresonant perturbation theory_ , _IEEE Transactions on Microwave Theory and Techniques_ 14 (1966) 70.
  * [16] C. Bohren and D.R. Huffman, _Absorption and Scattering of Light by Small Particles_ , Wiley Science Paperback Series (1998).
  * [17] S. Knirck, J. Schütte-Engel, A. Millar, J. Redondo, O. Reimann, A. Ringwald et al., _A first look on 3d effects in open axion haloscopes_ , _Journal of Cosmology and Astroparticle Physics_ 2019 (2019) 026.
  * [18] J. Ruze, _Antenna tolerance theory—a review_ , _Proceedings of the IEEE_ 54 (1966) 633.
  * [19] A.J. Millar, G.G. Raffelt, J. Redondo and F.D. Steffen, _Dielectric haloscopes to search for axion dark matter: theoretical foundations_ , _Journal of Cosmology and Astroparticle Physics_ 2017 (2017) 061–061.
  * [20] H. Wang, _Studies on dielectric properties of sapphire between 10ghz-40ghz_ , Master’s thesis, RWTH Aachen, November, 2022.
  * [21] T.M. collaboration, S. Knirck, J. Schütte-Engel, S. Beurthey, D. Breitmoser, A. Caldwell et al., _Simulating madmax in 3d: requirements for dielectric axion haloscopes_ , _Journal of Cosmology and Astroparticle Physics_ 2021 (2021) 034.
  * [22] M.A. Herráez, D.R. Burton, M.J. Lalor and M.A. Gdeisat, _Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path_ , _Appl. Opt._ 41 (2002) 7437.