††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Hyperfine structure and isotope shifts of the
${}^{1}P_{1}\leftarrow{}^{1}S_{0}$ transition in atomic zinc

David Röser  University of Bonn, Nussallee 12, 53115 Bonn, Germany  J. Eduardo
Padilla-Castillo  Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg
4-6, 14195 Berlin, Germany  Ben Ohayon  The Helen Diller Quantum Center,
Department of Physics, Technion-Israel Institute of Technology, Haifa,
3200003, Israel  Russell Thomas  Fritz-Haber-Institut der Max-Planck-
Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany  Stefan Truppe  Fritz-
Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin,
Germany  Centre for Cold Matter, Blackett Laboratory, Imperial College London,
London SW7 2AZ  Gerard Meijer  Fritz-Haber-Institut der Max-Planck-
Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany  Simon Stellmer
University of Bonn, Nussallee 12, 53115 Bonn, Germany  Sid C. Wright
<EMAIL_ADDRESS>Fritz-Haber-Institut der Max-Planck-Gesellschaft,
Faradayweg 4-6, 14195 Berlin, Germany

###### Abstract

We report absolute frequency, isotope shift, radiative lifetime and hyperfine
structure measurements of the ${}^{1}P_{1}\leftarrow$ ${}^{1}S_{0}$ (213.8 nm)
transition in Zn I using a cryogenic buffer gas beam. Laser-induced
fluorescence is collected with two orthogonally oriented detectors to take
advantage of differences in the emission pattern of the isotopes. This enables
clear distinction between isotopes whose resonances are otherwise unresolved,
and a measurement of the fermion hyperfine structure parameters,
$A(^{67}$Zn)$=20(2)$ MHz and $B(^{67}$Zn)$=10(5)$ MHz. We reference our
frequency measurements to an ultralow expansion cavity and achieve an
uncertainty at the level of 1 MHz, about 1 percent of the natural linewidth of
the transition.

††preprint: APS/123-QED

##  I Introduction

The alkaline-earth-metal (AEM) elements are identified by two valence
electrons and a $J=0$ electronic ground state. These two features give rise to
a number of unique properties. Firstly, the level structure decomposes into
singlet and triplet states, with broad transitions within each system and
narrow intercombination lines between them. Just as in the helium atom, the
lowest triplet states are metastable. Second, states with zero electronic spin
are free of hyperfine structure. In addition, bosonic isotopes have even
proton and even neutron numbers, leading to zero nuclear spin and absence of
hyperfine structure in all electronic states. These properties enable a wealth
of applications, including optical clocks [1], precision metrology [2],
quantum computing [3, 4, 5], and Rydberg physics [6, 7]. In recent years, AEM
elements have played a major role in the search for yet undiscovered scalar
gauge bosons through high-precision isotope shift spectroscopy [8], and
various studies with neutral AEM atoms have been presented on this topic [9,
10, 11, 12, 13, 14].

Alongside the AEM elements, and sharing these attractive properties, are the
so-called Group-IIB elements zinc, cadmium, and mercury. The broad singlet
${}^{1}P_{1}\leftarrow{}^{1}S_{0}$ transitions for these elements lie deep in
the ultraviolet range of the spectrum, with natural linewidths in excess of
100 MHz. They possess a multitude of bosonic and fermionic isotopes, with the
latter showing hyperfine structure. The resonance lines of the different
isotopes are convoluted and often cannot be resolved in conventional Doppler-
free spectroscopy. While Cd and Hg have already been employed for the
development of optical clocks [15, 16, 17], there is very modest work towards
this application with zinc thus far [18], mainly limited by the available
laser technology. The wider chain of radioactive zinc isotopes is of interest
for nuclear structure studies. For this reason, their isotope shifts have been
measured in the triplet manifold [19, 20], with new experiments ongoing [21,
22].

Here, we present high-resolution spectroscopy of the
${}^{1}P_{1}\leftarrow{}{}^{1}S_{0}$ transition near 213.8 nm in neutral zinc.
Experiments were conducted over a two week campaign in which an ultralow
expansion cavity and required deep ultraviolet optics (University of Bonn)
were transported to an atomic beam machine at the Fritz Haber Institute in
Berlin. Our measurements are based on laser-induced fluorescence of a
cryogenic beam of atoms extracted from a helium buffer gas cell. We employ a
two-detector method to clearly separate the contributions from the spin-zero
bosonic isotopes from the spin-5/2 fermionic isotope, enabling use of natural
abundance Zn. Isotope shifts and hyperfine interaction constants are
determined with an uncertainty of order 1 MHz. This method provides a
blueprint for measurements of hyperfine structure in strong optical
transitions, and a convenient and direct way to measure the true collection
solid angle of a fluorescence detector. Our approach can be readily adapted to
other species with several naturally occurring isotopes, e.g., Sn, Ni, and
applied in the study of radioactive nuclei.

##  II Experimental Setup

Figure 1(a) illustrates our experimental apparatus and laser system. We use a
cryogenic buffer gas source to produce a cold, slow atomic beam of zinc. The
atoms are produced by laser ablation of a solid Zn target (natural abundance),
are cooled in the cell by collisions with a He gas at a temperature of 3 K,
and exit the cell with a typical velocity of $140$ m/s along the $z$-axis. The
ablation laser is fired at a rate of 1 Hz which sets the repetition rate for
the experiments. At a distance 70 cm downstream of the cell exit, we excite
the ${}^{1}P_{1}\leftarrow{}^{1}S_{0}$ transition with a single probe laser
beam near 213.8 nm, which intersects the atomic beam perpendicularly. A
$2\times 2$ mm square aperture restricts the range of transverse velocities in
the atomic beam to below 1 m/s.

Figure 1: (a) Experimental setup for ${}^{1}P_{1}\leftarrow{}^{1}S_{0}$ laser
induced fluorescence spectroscopy of Zn, showing the cryogenic buffer gas
beam, and 213.8 nm laser system. We also show a side view of the detector
geometry, showing the two photomultipliers used, and the angles $\theta_{1}$
and $\theta_{2}$ relative to linear polarisation angle of the excitation
light. These angles determine the emission pattern observed at the two
detectors. (b) A typical time-of-flight fluorescence trace observed at the
64Zn resonance. The inset shows a zoom-in of the region $0<t<25$ ms.
Observation windows for the buffer gas cooled and thermal background
components of the signal are shown by the shaded bars. (c) Fluorescence
spectra for the observation windows in (b).

Continuous wave laser light is produced by twice frequency doubling the
infrared light of a Ti:Sa laser near 855.2 nm. Each frequency doubling stage
consists of an enhancement resonator containing a nonlinear crystal; to reach
213.8 nm from 427.6 nm we use beta Barium Borate ($\beta$-BBO). The 855.2 nm
light from the Ti:Sa laser is frequency stabilised either by referencing to a
commercial wavemeter (High Finesse WS8-10 calibrated with a temperature-
stabilised HeNe laser), or via an ultralow expansion (ULE) optical cavity
(Menlo ORC) with a measured free spectral range of 2.992184(30) GHz. We also
record the intermediate 427.6 nm light on the wavemeter, since this light is
immune from parasitic multi-mode content at the fundamental wavelength. The
wavemeter option offers an absolute accuracy of about 30 MHz when measuring
the 427.6 nm light, a resolution of about 1 MHz, and enables continuous
scanning over the entire spectrum. Scanning via the reference cavity reduces
the linewidth of the laser, and improves the linearity of the frequency axis.
To do this, light at the fundamental wavelength of the Ti:Sa laser is coupled
into a fiber phase modulator (EOM, Jenoptik) driven with two RF frequencies,
$\nu_{PDH}=18$ MHz and $\nu_{scan}\sim$1 GHz. The phase-modulated light
reflected from the cavity is collected on a fast photodiode, demodulated at
the frequency $\nu_{PDH}$ which produces a Pound-Drever-Hall (PDH) signal with
sharp zero crossing points when the laser frequency is at a cavity resonance
$\nu_{c}$, or at $\nu_{c}\pm\nu_{scan}$. We lock the laser to the latter, and
scan the laser frequency by varying $\nu_{scan}$. A camera is used to monitor
the light transmitted through the cavity and ensure locking to the TEM00 mode.
This locking scheme enables continuous scanning of the Ti:Sa laser frequency
up to one half of the cavity free spectral range, corresponding to 6 GHz at
the 213.8 nm detection wavelength.

At the atomic beam machine, we purify the laser polarisation with a polarising
beam cube, and control its linear polarisation angle relative to the direction
of the atomic beam with a $\lambda/2$ plate. The probe light propagates along
the $x$-axis and has a peak intensity $I=10^{-3}I_{sat}$, where $I_{sat}=\pi
hc\Gamma/(3\lambda^{3})=1.5~{}$W/cm2 is the two-level saturation intensity of
the transition. We estimate that an atom travelling through the maximum
intensity of the excitation light scatters five photons at resonance. The
resulting laser-induced fluorescence (LIF) is collected and imaged onto two
photomultiplier tubes (PMTs), whose photocurrents are delivered to separate
transimpedance amplifiers and recorded as time-of-flight traces. The two PMTs
are oriented to collect fluorescence emitted parallel and perpendicular to the
direction of the atomic beam, as shown in figure 1(a). The angle $\theta_{i}$
between the laser polarisation and the direction of detector $i$, illustrated
in the inset to the figure, determines the portion of the fluorescence
emission pattern collected by the two detectors. This enables discriminating
between fermionic and bosonic isotopes [23, 24, 13]. We record the laser power
after the machine with a calibrated optical power meter and compensate for
drifts in the probe intensity over a scan (typically $5-10$%).

Figure 1(b) shows a typical time-of-flight trace observed in detector 1 when
exciting the 64Zn resonance. The signal comprises an initial intense peak from
the buffer gas cooled atomic beam at roughly 5 ms, followed by an extended
tail which appears for several tens of ms later. The extended tail consists of
thermalised Zn atoms which leave the cell and collide with the vacuum walls;
it persists even when direct line of sight from the source to the detector is
blocked, and leads to a broad background signal in the fluorescence spectra,
whose Doppler width is consistent with the laboratory temperature. Example
spectra showing the two signal components are shown in Fig. 1(c).

##  III Analysis of spectral lineshapes

In the following, we discuss the lineshape models used for the fermionic and
bosonic isotopes, which are important in fitting the experimental spectra. We
hereon use $\nu_{L}$ to label the laser frequency and assume that the laser
linewidth is much less than the natural linewidth of the transition.

Boson lineshape -  The bosonic isotopes, all with nuclear spin $I_{N}=0$,
exhibit no hyperfine structure and the total angular momenta of the ground and
excited states are $F=0$ and $F^{\prime}=1$ respectively. The resonance line
of a boson $b$ can be simply described with the line function,

$S^{(b)}=\frac{\Gamma^{2}/4}{\Gamma^{2}/4+\Delta_{b}^{2}}[1-P_{2}(\cos\theta)g(\theta_{C})]\hskip
8.5359pt.$ (1)

Here, $\Gamma/(2\pi)$ is the Lorentzian linewidth of the transition,
$\Delta_{b}/(2\pi)=\nu_{L}-\nu_{b}$ is the detuning of the laser from the
resonance frequency $\nu_{b}$, and
$P_{2}(\cos\theta)=\frac{1}{2}(3\cos^{2}\theta-1)$ is the second Legendre
polynomial, with $\theta$ the angle between the detection direction and the
electric field of the linearly polarised excitation light. The factor
$g(\theta_{C})=\cos(\theta_{C})\cos^{2}(\theta_{C}/2)$ corrects for the effect
of the finite solid angle of the detection optics, with $\theta_{C}$ the half
angle of a circular collection lens. For $\theta=0$,
$\mathcal{S}^{(b)}\rightarrow 0$ as $\theta_{C}\rightarrow 0$, as would be
expected from the well-known Hertzian dipole radiation pattern. Importantly,
adjusting $\theta$ or $\theta_{C}$ changes the amplitude of the boson signal
observed at the detector.

Fermion lineshape -  There exists a single naturally abundant fermionic
isotope of Zn with nucleon number 67 and a nuclear spin $I_{N}=5/2$. The
nuclear spin couples with the electronic angular momentum $J$ to give total
angular momentum $F$, resulting in a single ${}^{1}S_{0},F=5/2$ hyperfine
level and three ${}^{1}P_{1},F^{\prime}$ excited levels with
$F^{\prime}=3/2,5/2,7/2$. These energy levels are shown in figure 2(a). We
assume the hyperfine energies $E(F)$ are given by

$E(F)=\frac{A}{2}C+B\frac{\frac{3}{4}C(C+1)-I_{N}(I_{N}+1)J(J+1)}{2I_{N}(2I_{N}-1)J(2J-1)}\hskip
2.84544pt,$ (2)

with $C=F(F+1)-I_{N}(I_{N}+1)-J(J+1)$. Here, $A=A(^{1}P_{1}$) is the
interaction between the electronic and nuclear angular momentum in the excited
state, and $B=B(^{1}P_{1})$ is the quadrupole interaction coefficient.

Following Brown et al. [25], the fluorescence spectrum of 67Zn, $S^{(f)}$ can
be separated into three terms:

$\begin{split}S^{(f)}=&\frac{\Gamma^{2}}{4}\Big{(}\mathcal{A}+[\mathcal{B}+\mathcal{C}]P_{2}(\cos\theta)g(\theta_{C})\Big{)}\hskip
8.5359pt,\\\
\mathcal{A}=&\frac{1}{9}(\frac{2}{\Gamma^{2}/4+\Delta_{3/2}^{2}}+\frac{3}{\Gamma^{2}/4+\Delta_{5/2}^{2}}+\frac{4}{\Gamma^{2}/4+\Delta_{7/2}^{2}})\hskip
5.69046pt,\\\
\mathcal{B}=&-\frac{1}{225}\frac{1}{\Gamma^{2}/4+\Delta_{3/2}^{2}}\\\
&-\frac{64}{525}\frac{1}{\Gamma^{2}/4+\Delta_{5/2}^{2}}-\frac{2}{21}\frac{1}{\Gamma^{2}/4+\Delta_{7/2}^{2}}\hskip
8.5359pt,\\\
\mathcal{C}=&\Big{[}-\frac{8}{45}\frac{1}{(\Gamma/2+i\Delta_{7/2})(\Gamma/2-i\Delta_{3/2})}\\\
&-\frac{6}{35}\frac{1}{(\Gamma/2+i\Delta_{7/2})(\Gamma/2-i\Delta_{5/2})}\\\
&-\frac{1}{25}\frac{1}{(\Gamma/2+i\Delta_{5/2})(\Gamma/2-i\Delta_{3/2})}\Big{]}+c.c.\end{split}$
(3)

Here, $\Delta_{F^{\prime}}/(2\pi)=\nu-\nu_{F^{\prime}}$ is the detuning of the
laser from the excited state with total angular momentum $F^{\prime}$; We
assumed all Zeeman sublevels in the ${}^{1}S_{0}$ state are equally populated
in the source and neglected optical pumping during the interaction with the
probe light.

Important for the experiments is the fact that when the hyperfine structure is
barely resolved, the emission pattern and hyperfine structure become strongly
coupled. This is illustrated by figure 2b, which shows simulated fluorescence
spectra along $\theta=0$ for different ratios of $A/\Gamma$. Each panel
compares equation (3) with the result when interference is removed from the
model, i.e. $\mathcal{C}$ is deliberately set to zero. The calculations show
that as $A/\Gamma\rightarrow 0$, interference between scattering paths is
destructive, leading to complete suppression of the fluorescence along this
direction. There is an intuitive explanation for this effect: when the
hyperfine interaction with the nucleus becomes negligible, the emission
pattern must converge to that of the (spin-less) bosonic isotopes. Conversely,
one can produce the reverse effect in the bosonic isotopes by deliberately
applying a magnetic field along $\theta=90^{\circ}$. This is the so-called
Hanle effect [26, 27] and, whilst understood for about a century, is often
overlooked. The behaviour illustrated in figure 2b shows that interference in
the emission pattern of barely resolved lines contains useful information
which can be used to constrain the hyperfine structure. The central spectrum
in the figure, where $A/\Gamma=0.2$, is near the value observed in the
experiments, and results in a total span of the ${}^{1}P_{1}$ levels of 1.2
$\Gamma$. This should largely prevent optical pumping to the $m_{F}=\pm 5/2$
states when driving the $F^{\prime}=3/2\leftarrow F=5/2$ transition, which
would be significant in the case of well-resolved lines.

Figure 2: Simulated fluorescence spectra for 67Zn. (a) Level scheme labelling
the total angular momenta $F,$ $F^{\prime}$ for the ground and excited states
respectively. (b) Simulated spectra for detection along $\theta=0$, for
different values of the ratio $A/\Gamma$ and with $B=0$. We show the results
with and without quantum interference included in the calculation. The sticks
above the spectra correspond to the energies of the levels in (a). As
$A/\Gamma$ approaches zero (negligible hyperfine interaction), the emission
pattern including interference converges to that of an ideal Hertzian dipole,
meaning the emission is zero along $\theta=0$.

Combined line function -  The total fit function used in this study is given
by,

$\begin{split}S^{(tot)}=&a_{67}S^{(f)}+\sum_{b}a_{b}S^{(b)}\\\
&+a_{bg}e^{-(\nu_{L}-\nu_{bg})^{2}/(2w_{bg}^{2})}\hskip 8.5359pt.\end{split}$
(4)

Here, $a_{67}$ and $a_{b}$ represent the relative abundance of the fermionic
and bosonic isotopes respectively. The final term in equation (4) approximates
the residual thermal background in the spectrum, whose amplitude $a_{bg}$ is
typically 5 to 10 percent of the 64Zn resonance peak. The centre frequency
$\nu_{bg}$ and width parameter $w_{bg}$ can be either fitted as free
parameters, or introduced as fixed parameters by first fitting the data at
late arrival times when only the thermal background component is present. The
fitted values for the isotope shifts in these two cases are consistent within
the statistical error of the fits.

##  IV Results

###  IV.1 Determination of the fermion hyperfine structure by a two-detector
method

Figures 3(a) and (b) show two sets of spectra obtained using the High Finesse
wavemeter as a frequency reference. The data constitutes two separate scans
where the input polarisation of the laser is along the $y$-axis (panel (a))
and along the $z$-axis (panel (b)), and for each panel we show the
fluorescence spectrum recorded by the two detectors $1$ and $2$. For clarity,
each is labelled with a schematic showing the laser polarisation, the detector
orientation and the dominant emission pattern for the bosonic isotopes. The
different emission pattern of the fermionic 67Zn isotope (relative natural
abundance $4.1\%$) dramatically increases its visibility in detector $i$ when
$\theta_{i}=0$. We show the fitted fermion lineshape with a black dashed line
in each panel to illustrate this effect.

Figure 3: Polarisation sensitive fluorescence detection of Zn isotopes. Each
spectrum is labelled by the probe laser polarisation and detector
configuration. The relative intensities of the boson and fermionic resonances
are strong functions of the angle $\theta$ between the laser polarisation and
the detector direction, and the solid angle of the collection optics. Blue
lines show experimental data, red solid lines are fits as described in the
text, and black dashed lines show the fitted fermion lineshape. Underneath
each spectrum the residuals are shown in a separate plot (‘Res’). (a) Laser
polarisation along the $y$-axis. (b) Laser polarisation along the $z$-axis.

We use these spectra to determine solid angles of the collection optics and
hyperfine structure of the 67Zn isotopes. The four spectra in figure 3 were
fitted as a single dataset, fixing the detection angles $\theta_{i}$ to their
values in the experiment, and enforcing the natural abundance of Zn isotopes
[28]. This fixes the relative peak heights in each spectrum so that the
detector solid angles $\theta_{C,1},\theta_{C,2}$, and the hyperfine structure
constants $A$ and $B$ of the fermionic isotope, can be determined. All
resonance frequencies $\nu_{b}$ for all bosons $b$, $\nu_{1/2,3/2,5/2}$ for
the fermionic 67Zn isotope, and a common Lorentzian linewidth $\Gamma$ are
shared fit parameters between the datasets. From this data, we conclude
$\theta_{C,1}=0.281\pm 0.005$, $\theta_{C,2}=0.145\pm 0.005$ radians. The
uncertainties are the range of values obtained when fitting the data with
various reasonable assumptions, such as fixing the values of $w_{bg}$ and
$\nu_{bg}$ in the fit function using the signal at late arrival times. The
value of $\theta_{C,2}$ is very close to the half angle subtended by the
collection lens at the fluorescence region, $0.156(5)$ radians. The value of
$\theta_{C,1}$ is significantly below the half angle subtended by its in-
vacuum collection lens, $0.43(1)$ radians, and consistent with this lens being
placed about $5$ mm too close to the atomic beam, a result of incorrectly
extrapolating the focal length from the visible to the deep ultraviolet.

For the hyperfine interaction parameters, we obtain $A(^{67}$Zn$)=20\pm 2$ MHz
and $B(^{67}$Zn$)=10\pm 5$ MHz. Fitting the data with $B=0$ returns $A=21$ MHz
but noticeably reduces the goodness of the fit near the 67Zn peak. Fitting to
a model which ignores interference may be done by simply setting
$\mathcal{C}=0$ in equation (3); this gave the best fit values $A=9.5\pm 1.8$
MHz, $B=0.6\pm 2.2$ MHz, $\theta_{C,1}=0.37$ and $\theta_{C,2}=0.20$, and line
centres consistent with the full interference model. In this case the fitted
value of $\theta_{C,2}$ is unphysically large, and the fit residuals clearly
indicate that only the interference model can adequately describe the signal
observed in both detectors.

###  IV.2 High resolution measurements with the cavity

Having constrained the fermion hyperfine structure and solid angle of the
collection optics, we proceeded to scan the laser via the ULE reference cavity
to more accurately determine the isotope shifts. Figure 4 shows spectra
obtained when scanning $\nu_{scan}$ with the laser locked to the ULE cavity,
and with the probe laser horizontally polarised. The scan rate corresponds to
approximately 0.8 MHz/s for the 213.8 nm probe light, where the frequency
$\nu_{scan}$ was measured near the time of the ablation laser pulse, and then
stepped discretely after each measurement. By happenstance, the 66Zn resonance
appeared almost exactly at the midpoint between two cavity resonances, where
the locking method fails. We therefore frequency-shifted the Ti:Sa laser light
by $90$ MHz before delivery to the cavity with an acoustic optic modulator,
moving the unstable lock point by 360 MHz in the deep ultraviolet. The upper
(lower) dataset in the figure is taken with (without) the frequency shifting
method applied, on the same day but ablating different spots on the Zn target,
which enabled measuring all isotopes. We fit the two spectra as a single
dataset with shared resonance line positions in a Monte Carlo routine, where
the values of $\theta_{C,1},\theta_{C,2},A$ and $B$ are drawn from uniform
distributions whose ranges are given by the limits constrained in section
IV.1. Enforcing the relative natural abundance of Zn in the fits leads to
small changes in the best fit values compared to allowing the line intensities
to float, and we include this when estimating the uncertainties. We combine
the best-fit values and errors for the isotope shifts to give a weighted mean
and statistical error for these parameters, and assume a $1$ MHz systematic
frequency uncertainty which derives from the $\sim 200$ kHz uncertainty of
$\nu_{scan}$ and the frequency shifting AOM, considering the two successive
stages of frequency doubling. Doppler shifts due to slight misalignment of the
probe laser light contribute an uncertainty $\sim 2$ MHz to the absolute
resonance frequencies, and negligibly to the isotope shifts. Recoil from
absorption of the probe laser light leads to a Doppler shift of roughly 0.6
MHz across the detection volume, and the differential shift across the range
of isotopes is an order of magnitude smaller. We neglect this contribution to
the isotope shift uncertainty. The ambient magnetic field in the detector was
measured as below 0.3 Gauss, corresponding to an upper bound to the line shape
broadening of 0.5 MHz.

Figure 4: Fluorescence spectra taken with the probe laser locked to the ULE
cavity. The shaded box shows the region near the midpoint of the cavity
resonances in which the lock fails. In the upper spectrum, shown with a
deliberate offset on the $y$-axis, the light delivered to the cavity was first
frequency-shifted by 90 MHz. This enables a continuous scan over the 66Zn
resonance. The two spectra are fitted as a single dataset to recover the
isotope shifts as discussed in the text. Dashed lines show the fit functions
where data is not present for each spectrum. The plot underneath shows the
residual from the fit function for the two datasets.

##  V Discussion

Table 1 summarises the results of our measurements, and compares them to the
available literature values. Our final values for the isotope shifts are
presented relative to the 64Zn resonance, since this isotope is of highest
abundance and its line centre has the smallest statistical uncertainty. We
combine measurements from the cavity and spectra taken using the wavemeter,
and include a systematic frequency error of $2.1$ MHz for the wavemeter
values, which derives from directly comparing frequency intervals measured by
the cavity scan method with the wavemeter. Our values are two orders of
magnitude more precise than previous measurements, also given in the table.
Our value of $A(^{67}$Zn$)$ is consistent with the value measured by Kowalski
and Träger [29], by level crossing spectroscopy of enriched 67Zn. However,
this study was unable to experimentally constrain the value of $B(^{67}$Zn$)$
and we therefore recommend the values from our measurements.

Table 1:  Summary of the results obtained for the ${}^{1}P_{1}\leftarrow{}^{1}S_{0}$ transition in Zn. Hyperfine constants and the radiative lifetime refer to the excited state. Results are given in MHz unless otherwise stated. CG = centre of gravity.  | This work | Ref.[30] | Ref.[31] | Ref.[29] | Ref.[32] | Ref.[33] |
---|---|---|---|---|---|---|---
$\nu_{66}-\nu_{64}$ | 525.0(3.0) | $480(60)$ | $540(60)$ |  |  |  |
$\nu_{67}^{\mathrm{(CG)}}-\nu_{64}$ | 835(4) | - | - |  |  |  |
$\nu_{68}-\nu_{64}$ | 1039.8(1.7) | $989(60)$ | $960(85)$ |  |  |  |
$\nu_{70}-\nu_{64}$ | 1495(4) | - | - |  |  |  |
$A(^{67}$Zn$)$ | 20(2) |  |  | $17.7(5)$ |  |  |
$B(^{67}$Zn$)$ | 10(5) |  |  | - |  |  |
$\nu(^{1}P_{1}-{}^{1}S_{0})$/cm-1 | $46745.407(2)$ |  |  |  | $46745.404(2)$ |  |
$\tau$/ns | 1.440(18) |  |  |  |  | 1.40(3) |

The absolute frequency of the 64Zn resonance measured through our experiments
is $1,401,391.66(6)$ GHz ($46745.394(2)$ cm-1). The isotope-averaged line
centre, $\nu(^{1}P_{1}-{}^{1}S_{0})$, is computed as the average of the line
centres weighted by isotopic abundance and given in the table. The uncertainty
in our value is dominated by the wavemeter accuracy specified by the
manufacturer. We find excellent agreement with the results of (isotope-
unresolved) hollow cathode lamp measurements presented in Ref. [32].

Our best fit value of the Lorentzian linewidth is $\Gamma/(2\pi)=110.5(1.4)$
MHz, where the error derives from the standard deviation of the Monte Carlo
fitted values combined with the frequency uncertainty from the cavity scanning
method. Fitting with a Voigt lineshape did not change the value of $\Gamma$
within the uncertainty of the fit. The radiative lifetime $\tau=1/\Gamma$
given in the table is consistent with the weighted average of 5 measurements
collated by Doidge [33], and is a factor of two more precise than previous
measurements.

King plot \- We combine our isotope shift results with values reported for the
${}^{3}P_{1}\leftarrow{}^{1}S_{0}$ intercombination transition [34] on a King
plot as follows. We calculate the reduced isotope shifts
$\delta\bar{\nu}^{A,A^{\prime}}=\delta\nu/\mu^{A,A^{\prime}}$ with
$\mu^{A,A^{\prime}}$ the inverse nuclear mass difference between isotopes $A$
and $A^{\prime}$, and present the data in figure 5. The data is fitted to a
linear relationship according to the recipe described in Ref. [35]. Briefly,
we first define a mixing matrix to shuffle the 308 nm reduced isotope shifts
so that they are referred to 64Zn. We then calculate the covariance matrices,
taking into account the mixing matrix and the reported errors. The most-
probable values of the adjusted parameters, the intercept and slope, are found
by minimizing a generalized $\chi^{2}$ test statistic. To assign confidence
intervals to the fitted parameters, we perform a Monte-Carlo estimation
procedure of repeated measurements drawn from a normal distribution centred at
the most probable fitted values. The most probable value and $68\%$ confidence
interval of the fitted line is plotted in figure 5.

Figure 5: A King plot of the ${}^{1}P_{1}\leftarrow{}^{1}S_{0}$ (213.8 nm,
this work) and ${}^{3}P_{1}\leftarrow{}^{1}S_{0}$ (308 nm, ref. [34])
transitions in Zn. The black solid line shows a linear fit to the data as
discussed in the text. Red dashed lines indicate the $68\%$ confidence
interval of the fit.

An additional hyperfine interaction between the different electronic states in
Zn would additionally shift the gravity centre of the 67Zn resonance lines,
$\nu_{67}^{(\mathrm{CG})}$, relative to the bosons, and is often referred to
as an “off-diagonal” hyperfine interaction. Such shifts should be detectable
in a King plot as deviations of the fermionic isotopes from linear fits of the
bosonic isotope data (see for example ref. [9]). To test for this, we repeat
the fitting procedure without the (64,67) pair, and calculate its predicted
values in each iteration of the Monte-Carlo procedure to obtain its
distribution. The resulting $68\%$ confidence interval for the frequency
difference $\nu_{67}^{(\mathrm{CG})}-\nu_{64}$ is from 818 MHz to 834 MHz,
which agrees with our measured value (835(4) MHz) within its uncertainty. We
infer that the difference in off-diagonal hyperfine shifts of the 67Zn
resonances, for this pair of transitions, is less than $10\,$MHz. This closely
follows results of isotope shift measurements for the lowest lying, and
analogous, transitions in cadmium [13], another Group-IIB element. It may be
of interest considering recently observed strong hyperfine mixing effects for
higher-lying transitions in zinc in the CRIS experiment [22].

##  VI Summary and outlook

We have reported isotope shifts, radiative lifetime and hyperfine structure
measurements for the ${}^{1}P_{1}\leftarrow{}^{1}S_{0}$ transition in neutral
Zn by cw laser-induced fluorescence spectroscopy of an atomic beam. Our
measurements considerably improve upon the published literature for this
transition, and contribute to the study of the 67Zn nucleus, where unexpected
isotope shifts have recently been observed in collinear laser spectroscopy at
the ISOLDE facility [22]. With its multitude of spin-zero bosonic isotopes and
various narrow optical transitions, zinc is a candidate for further isotope
shift spectroscopy at the sub-kHz level.

The two-detector method and analysis procedure described here has several
advantages. First, it enables reliably extracting hyperfine parameters from
barely resolved peaks. Second, it enables tuning of the fermion contribution
to spectra of mixed isotopes within a single measurement run. This approach
allows disentangling otherwise overlapping lines and in the case of atomic Zn
studied here, enables a measurement of the isotope shifts and hyperfine
structure at the $\sim$1 MHz level. The approach can readily be adopted to
other elements which feature broad transitions and many isotopes, and may be
particularly beneficial in accelerator-based experiments, where measurements
typically come with a large overhead.

###### Acknowledgements.

We thank Sebastian Kray and the mechanical workshop of the Fritz Haber
Institute for expert technical assistance. S.W. thanks Clara Bachorz for
critical reading of the manuscript. We gratefully acknowledge financial
support from the European Research Council (ERC) under the European Union’s
Horizon 2020 Research and Innovation Programme (CoMoFun, Grant Agreement No.
949119, S.T.; “quMercury”, GA No. 757386, S.S.), from the European Commission
through project 101080164 “UVQuanT” (S.T. and S.S.), and from Deutsche
Forschungsgemeinschaft (DFG) through grants 414709674 and 496941189 and
through the Cluster of Excellence ML4Q (EXC 2004/1 - 390534769). B.O. is
thankful for the support of the Council for Higher Education Program for
Hiring Outstanding Faculty Members in Quantum Science and Technology.

## References

  * Ludlow  _et al._ [2015] A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, Rev. Mod. Phys. 87, 637 (2015).
  * Tarallo  _et al._ [2014] M. G. Tarallo, T. Mazzoni, N. Poli, D. V. Sutyrin, X. Zhang, and G. M. Tino, Phys. Rev. Lett. 113, 023005 (2014).
  * Daley  _et al._ [2008] A. J. Daley, M. M. Boyd, J. Ye, and P. Zoller, Phys. Rev. Lett. 101, 170504 (2008).
  * Gorshkov  _et al._ [2009] A. V. Gorshkov, A. M. Rey, A. J. Daley, M. M. Boyd, J. Ye, P. Zoller, and M. D. Lukin, Phys. Rev. Lett. 102, 110503 (2009).
  * Saffman [2020] M. Saffman, J. Phys. B: At. Mol. Opt. Phys 49, 202001 (2020).
  * Simien  _et al._ [2004] C. E. Simien, Y. C. Chen, P. Gupta, S. Laha, Y. N. Martinez, P. G. Mickelson, S. B. Nagel, and T. C. Killian, Phys. Rev. Lett. 92, 143001 (2004).
  * Browaeys and Layahe [2020] A. Browaeys and T. Layahe, Nature Physics 16, 132 (2020).
  * Berengut  _et al._ [2018] J. C. Berengut, D. Budker, C. Delaunay, V. V. Flambaum, C. Frugiuele, E. Fuchs, C. Grojean, R. Harnik, R. Ozeri, G. Perez, and Y. Soreq, Phys. Rev. Lett. 120, 091801 (2018).
  * Miyake  _et al._ [2019] H. Miyake, N. C. Pisenti, P. K. Elgee, A. Sitaram, and G. K. Campbell, Phys. Rev. Res. 1, 033113 (2019).
  * Witkowski  _et al._ [2019] M. Witkowski, G. Kowzan, R. Munoz-Rodriguez, R. Ciuryło, P. S. Żuchowski, P. Masłowski, and M. Zawada, Opt. Express 27, 11069 (2019).
  * Figueroa  _et al._ [2022] N. L. Figueroa, J. C. Berengut, V. A. Dzuba, V. V. Flambaum, D. Budker, and D. Antypas, Phys. Rev. Lett. 128, 073001 (2022).
  * Ono  _et al._ [2022] K. Ono, Y. Saito, T. Ishiyama, T. Higomoto, T. Takano, Y. Takasu, Y. Yamamoto, M. Tanaka, and Y. Takahashi, Phys. Rev. X 12, 021033 (2022).
  * Hofsäss  _et al._ [2023] S. Hofsäss, J. E. Padilla-Castillo, S. C. Wright, S. Kray, R. Thomas, B. G. Sartakov, B. Ohayon, G. Meijer, and S. Truppe, Phys. Rev. Res. 5, 013043 (2023).
  * Ohayon  _et al._ [2022] B. Ohayon, S. Hofsäss, J. E. Padilla-Castillo, S. C. Wright, G. Meijer, S. Truppe, K. Gibble, and B. K. Sahoo, New Journal of Physics 24, 123040 (2022).
  * Yamaguchi  _et al._ [2019] A. Yamaguchi, M. S. Safronova, K. Gibble, and H. Katori, Phys. Rev. Lett. 123, 113201 (2019).
  * Sarlo  _et al._ [2016] L. D. Sarlo, M. Favier, R. Tyumenev, and S. Bize, Journal of Physics: Conference Series 723, 012017 (2016).
  * Tyumenev  _et al._ [2016] R. Tyumenev, M. Favier, S. Bilicki, E. Bookjans, R. L. Targat, J. Lodewyck, D. Nicolodi, Y. L. Coq, M. Abgrall, J. Guéna, L. D. Sarlo, and S. Bize, New Journal of Physics 18, 113002 (2016).
  * Büki  _et al._ [2021] M. Büki, D. Röser, and S. Stellmer, Applied Optics 60, 9915 (2021).
  * Wraith  _et al._ [2017] C. Wraith, X. Yang, L. Xie, C. Babcock, J. Bieroń, J. Billowes, M. Bissell, K. Blaum, B. Cheal, L. Filippin, R. Garcia Ruiz, W. Gins, L. Grob, G. Gaigalas, M. Godefroid, C. Gorges, H. Heylen, M. Honma, P. Jönsson, S. Kaufmann, M. Kowalska, J. Krämer, S. Malbrunot-Ettenauer, R. Neugart, G. Neyens, W. Nörtershäuser, F. Nowacki, T. Otsuka, J. Papuga, R. Sánchez, Y. Tsunoda, and D. Yordanov, Physics Letters B 771, 385 (2017).
  * Xie  _et al._ [2019] L. Xie, X. Yang, C. Wraith, C. Babcock, J. Bieroń, J. Billowes, M. Bissell, K. Blaum, B. Cheal, L. Filippin, K. Flanagan, R. Garcia Ruiz, W. Gins, G. Gaigalas, M. Godefroid, C. Gorges, L. Grob, H. Heylen, P. Jönsson, S. Kaufmann, M. Kowalska, J. Krämer, S. Malbrunot-Ettenauer, R. Neugart, G. Neyens, W. Nörtershäuser, T. Otsuka, J. Papuga, R. Sánchez, Y. Tsunoda, and D. Yordanov, Physics Letters B 797, 134805 (2019).
  * Yang  _et al._ [2020] X. Yang, T. Cocolios, and S. Geldhof, Probing the magicity and shell evolution in the vicinity of $n=50$ with high-resolution laser spectroscopy of 81,82Zn isotopes (2020).
  * Liu  _et al._ [2022] Y. Liu, X. Yang, S. Bai, J. Reilly, T. Cocolios, K. Flanagan, R. G. Ruiz, F. Gustafsson, J. Li, M. Ma, _et al._ , ISOLDE workshop and users meeting (2022).
  * Rasmussen  _et al._ [1974] W. Rasmussen, R. Schieder, and H. Walther, Optics Communications 12, 315 (1974).
  * Baird  _et al._ [1979] P. E. G. Baird, R. J. Brambley, K. Burnett, D. N. Stacey, D. M. Warrington, and G. K. Woodgate, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 365, 567 (1979).
  * Brown  _et al._ [2013] R. C. Brown, S. Wu, J. V. Porto, C. J. Sansonetti, C. E. Simien, S. M. Brewer, J. N. Tan, and J. D. Gillaspy, Phys. Rev. A 87, 032504 (2013).
  * Hanle [1923] W. Hanle, Naturwissenschaften 11, 690 (1923).
  * Mitchell and Zemansky [1961] A. C. G. Mitchell and M. W. Zemansky,  _Resonance Radiation and Excited Atoms_ (Cambridge University Press, London, 1961) Chap. V.
  * Rosman and Taylor [1998] K. J. R. Rosman and P. D. P. Taylor, Pure and Applied Chemistry 70, 217 (1998).
  * Kowalski and Träger [1976] J. Kowalski and F. Träger, Zeitschrift für Physik A Atoms and Nuclei 278, 1 (1976).
  * Crawford  _et al._ [1950] M. Crawford, A. Schawlow, W. Gray, and F. Kelly, Canadian Journal of Research 28, 138 (1950).
  * Hately and Littlefield [1958] G. Hately and T. Littlefield, JOSA 48, 851 (1958).
  * Gullberg and Litzén [2000] D. Gullberg and U. Litzén, Physica Scripta 61, 652 (2000).
  * Doidge [1995] P. Doidge, Spectrochimica Acta Part B: Atomic Spectroscopy 50, 209 (1995).
  * Campbell  _et al._ [1997] P. Campbell, J. Billowes, and I. S. Grant, Journal of Physics B: Atomic, Molecular and Optical Physics 30, 2351 (1997).
  * Sahoo and Ohayon [2023] B. Sahoo and B. Ohayon, arXiv preprint arXiv:2307.09092 (2023).


# Single-epoch and Differential Astrometric Microlensing of Quasars

R. Forés-Toribio Departamento de Astronomía y Astrofísica, Universidad de
Valencia, E-46100 Burjassot, Valencia, Spain  Observatorio Astronómico,
Universidad de Valencia, E-46980 Paterna, Valencia, Spain  E. Mediavilla
Instituto de Astrofísica de Canarias, Vía Láctea S/N, La Laguna, E-38200,
Tenerife, Spain  Departamento de Astrofísica, Universidad de la Laguna, La
Laguna, E-38200, Tenerife, Spain  J. A. Muñoz Departamento de Astronomía y
Astrofísica, Universidad de Valencia, E-46100 Burjassot, Valencia, Spain
Observatorio Astronómico, Universidad de Valencia, E-46980 Paterna, Valencia,
Spain  J. Jiménez-Vicente Departamento de Física Teórica y del Cosmos,
Universidad de Granada, Campus de Fuentenueva, 18071 Granada, Spain  Instituto
Carlos I de Física Teórica y Computacional, Universidad de Granada, 18071
Granada, Spain  C. Fian Departamento de Astronomía y Astrofísica, Universidad
de Valencia, E-46100 Burjassot, Valencia, Spain  C. del Burgo Instituto de
Astrofísica de Canarias, Vía Láctea S/N, La Laguna, E-38200, Tenerife, Spain
Departamento de Astrofísica, Universidad de la Laguna, La Laguna, E-38200,
Tenerife, Spain Raquel Forés-Toribio<EMAIL_ADDRESS>
###### Abstract

We propose and discuss a new experimental approach to measure the centroid
shift induced by gravitational microlensing in the images of lensed quasars
(astrometric microlensing). Our strategy is based on taking the photocenter of
a region in the quasar large enough as to be insensitive to microlensing as
reference to measure the centroid displacement of the continuum. In this way,
single-epoch measurements of astrometric microlensing can be performed. Using
numerical simulations, we show that, indeed, the centroid shift monotonically
decreases as the size of the emitting region increases, and only for
relatively large regions, like the broad line region (BLR), does the centroid
shift become negligible. This opens interesting possibilities to study the
stratification of the different emitters in the accretion disk and the BLR. We
estimate the amplitude of the centroid shifts for 79 gravitationally lensed
images and study more thoroughly the special cases Q2237+030 A, RXJ1131$-$1231
A, PG1115+080 A2 and SDSS J1004+4112 A. We propose to use spectro-astrometry
to simultaneously obtain the photocenters of the continuum and of different
emission line regions since, with the precision of forthcoming instruments,
astrometric microlensing by $\sim 1M_{\odot}$ mass microlenses may be detected
in many quasar lensed images. When we consider more massive micro/millilenses,
$M\gtrsim 10M_{\odot}$, often proposed as the constituents of dark matter, the
BLR becomes sensitive to microlensing and can no longer be used as a
positional reference to measure centroid shifts. Differential microlensing
between the images of a lensed quasar along several epochs should be used
instead.

Astronomical techniques (1684), Gravitational microlensing (672), Astrometric
microlensing effect (2140), Spectroscopy (1558), Quasars (1319)

##  1 Introduction

By studying the effect of gravitational microlensing on lensed quasars (Chang
& Refsdal, 1979; Wambsganss, 2006), it is possible to infer properties
concerning both the microlenses and the quasars: whether there are compact
objects within the lens and, if so, what is their mass distribution (see,
e.g., Kochanek, 2004; Mediavilla et al., 2009; Jiménez-Vicente & Mediavilla,
2019; Awad et al., 2023), and the size and brightness profile of the quasars
(see, e.g., Eigenbrod et al., 2008; Motta et al., 2012; Morgan et al., 2012;
Jiménez-Vicente et al., 2014; Muñoz et al., 2016; Fian et al., 2023). In order
to draw conclusions about these properties, the classical approach is to
measure temporal changes in the brightness of the lensed images.

Microlensing-induced changes in brightness are caused by the splitting of a
macroimage in several deformed and irregularly disposed microimages that
cannot be resolved with present-day telescopes, but affects the centroid of
the observed macroimages. Thus, present and future improvements in astrometric
instrumentation may allow to study the astrometric microlensing effect. The
main idea is to measure the photocenter of the images and study their shifts,
instead of their temporal variability in brightness. Lewis & Ibata (1998)
first proposed this type of study in the context of galaxy-quasar
microlensing. Honma & Kurayama (2002); Salata & Zhdanov (2003); Treyer &
Wambsganss (2004); Chen et al. (2007); Popović & Simić (2013) continued to
explore the astrometric microlensing and millilensing of quasars in different
scenarios.

A possible baseline for the displacements can be a theoretical reference
obtained from a macromodel of the lens system. However, the models can have
systematic positional errors of around several milliarcseconds (mas), see
e.g., Chen et al. (2007); Sluse et al. (2012). Another procedure proposed by
Treyer & Wambsganss (2004) consists of measuring the relative shift between
multiple images of a lensed quasar. The main drawback of this approach is that
it relies on measurements over months or years as the classical method to
study microlensing based on changes in magnitude.

Here, we are going to propose and study a third approach. It consists of
choosing a larger component of the quasar as the baseline. The principal
hypothesis is that the larger component is less affected by the microlensing
effect since the distortions in the final image produced by the many stars
back-mapped to the region covered by the source at the source plane are
compensated and diluted. Thus, the first step to apply the single-epoch method
is to show that the displacement produced by the stars decreases with
increasing size. The most important advance of this method is that it can be
applied to single-epoch measurements since their reference does not depend on
previous measurements.

Another basic information needed to asses the feasibility of astrometric
microlensing measurements in different scenarios is the amplitude of the
centroid shifts. To obtain an order of magnitude estimate we can consider that
for typical fractions of mass in stars ($\alpha\sim 10-20$ %) and sizes of the
quasar source (from $1$ to $5$ light-days), the rays coming from the source
will be mainly affected by a single or a few stars (low optical depth
approximation) and, consequently, we can expect that the images of the source
are going to be asymmetrically distributed with a separation of the order of
an Einstein radius. This separation is going to be enlarged by the
macromagnification induced by the total distribution of matter (microlenses
and smooth component) and the external shear. Thus, the centroid shift along
each principal axis would be of the order of an Einstein radius multiplied by
some linear magnification (to be determined with simulations). Then, if the
linear magnification is $\mu_{\pm}$ along each principal axis (as defined in
§2), respectively, we can expect a centroid shift of around
$\sqrt{\mu_{+}^{2}+\mu_{-}^{2}}$ Einstein radii. For a $0.3M_{\odot}$ star the
Einstein ring of a typical lens (source at redshift 2 and lens at redshift
0.5), is $10$ light-days at the source plane, which corresponds to 1
microarcsecond, $\mu$as. Taking into account that the Einstein radius is
proportional to $\sqrt{M}$, we can expect centroid shifts of, roughly,

$\Delta C\sim\sqrt{\mu_{+}^{2}+\mu_{-}^{2}}\sqrt{M/0.3M_{\odot}}\,\rm\mu as.$
(1)

Hence, centroid shifts of a few tens of $\mu$as, measurable with existent and
planned instrumentation, are expected. If we consider microlenses more massive
than the stars, the centroid shifts can be as large as hundreds of $\mu$as
(for LIGO-Virgo-KAGRA detected black holes) or several mas (for primordial
massive black holes, PMBH, or primordial black hole clusters with masses
larger than $10^{3}M_{\odot}$). However, for lenses masses above the stellar
mass range, the original reference can also experiment centroid shifts and we
should either look for a larger region unaffected for these mass ranges (like
the narrow line region, NLR) or, alternatively, measure the differential
centroid shifts between lensed quasar images in several epochs.

The article is organized as follows. In §2 we describe the methodology of the
numerical simulations. The dependence of the centroid shift with several
variables for a generic lens system is studied in §3, and §4 is devoted to a
quantitative study of the effect for specific lens systems. Possible
experimental measurements in different scenarios are discussed in §5, and the
main conclusions are summarized in §6.

##  2 Numerical methods and simulations

We perform all simulations with an adaptation of the Fast Multipole Method –
Inverse Polygon Mapping111https://gloton.ugr.es/microlensing/ (FMM-IPM)
developed by Jiménez-Vicente & Mediavilla (2022). We calculate the image of a
source by inverse ray shooting. As there are many lenses and rays involved, we
use the FMM to effectively calculate the ray’s deflections.

We model the sources as uniform circles sampled with at least $10^{5}$ pixels
regardless of the source size. Similarly, the resolution in the image plane is
set so that all images are sampled with at least N=$10^{5}$ pixels in absence
of microlensing. The area of the imaged source with radius $r_{s}$ in units of
the Einstein radius, $R_{E}$, in an environment of macromagnification $\mu$ is
$A=\mu\pi r_{s}^{2}$. The resolution (res=$R_{E}/$pixel) that makes this area
sampled by at least N pixels is $A/\text{res}^{2}\geq N$ or equivalently
$\text{res}\leq\sqrt{\mu\pi r_{s}^{2}/N}$. In this way, all images are, on
average, sampled with the same number of pixels. Given that the stochastic
deflection of the stars is $\sqrt{\kappa_{\ast}}R_{E}$ (Katz et al., 1986; Dai
& Pascale, 2021), the image plane size needs to be enlarged beyond the
magnification in each principal axis, i.e., $\mu_{-}=1/(1-\kappa-\gamma)$ and
$\mu_{+}=1/(1-\kappa+\gamma)$, to ensure that the majority of the microimages
are recovered. Zheng et al. (2022) defined a “protective border” of
$10\sqrt{\kappa_{\ast}}R_{E}$ per side, which ensures that at least 98% of the
light rays fall into the studied region of the source plane. In our case, the
size of the image plane is set to the magnification in each principal axis
multiplied by an “allowance” factor. This factor is determined after a
preliminary study for each individual case for which the image centroid is
stabilized and always enlarges the image plane area more than
$10\sqrt{\kappa_{\ast}}R_{E}$ per side. Generally, we need to extend the image
plane beyond that border limit because accurate centroid determinations
require the retrieval of microimages located far from the image plane center.

The stars have the same mass and are randomly distributed in a circle on the
image plane. The number of stars is fixed in the simulations for each study
case (i.e., Sections 3.1, 3.2, 4.1, 4.2, 4.3, and 4.4) to keep the same
Poissonian uncertainty. The number of stars needed to stabilize the centroid
shift in the worst-case scenario of each section is the one adopted for the
rest of the simulations. To achieve sufficient statistical significance, all
results are reported after simulating 1000 realizations for each set of
parameters.

We use as an input the convergence $\kappa$, shear $\gamma$, fraction in stars
$\alpha=\kappa_{\ast}/\kappa$, as well as the stellar distribution and the
source radius $r_{s}$. We calculate the image of our circular uniform source
and save the pixels (j1,j2) in the image plane where there is an image (i.e.
the inverse ray hits the source). For the rest of the image plane we have
I(j1,j2)=0. From this image matrix $I$, we can compute the image centroid in
image plane pixel coordinates as:

$j_{\text{cent},k}=\frac{\sum\limits_{j_{1}=1}^{n_{1}}\sum\limits_{j_{2}=1}^{n_{2}}I(j_{1},j_{2})j_{k}}{\sum\limits_{j_{1}=1}^{n_{1}}\sum\limits_{j_{2}=1}^{n_{2}}I(j_{1},j_{2})},$
(2)

with $k$=1, 2 for the principal axes and $n_{1}$, $n_{2}$ are the number of
pixels in each image plane axis. The centroid in pixel coordinates
$(j_{\text{cent},1},j_{\text{cent},2})$ is transformed into physical
coordinates $(\Delta C_{1},\Delta C_{2})$, whose origin is at the center of
the source.

##  3 Results for a generic case

To better grasp the qualitative phenomenology of astrometric microlensing we
start studying a generic case fixing as few parameters of the lens system as
possible. The idea is to express all the computations in terms of the Einstein
radius, which encompasses the geometry of the system (i.e. the redshifts of
the source and the lens, $z_{s}$ and $z_{l}$, respectively) and the mass of
the microlenses, $M$.

Figure 1: Centroid shift dependence of a 1$R_{E}$ source for different values
of convergence and shear while keeping the convergence in microlenses fixed to
$\kappa_{\ast}$=0.09 with respect to the linear macromagnification,
$\mu_{\text{lin}}$.

Figure 2: Centroid shift dependence of a 1$R_{E}$ source in each principal
axis ($\Delta C_{1}$ and $\Delta C_{2}$) for different values of convergence
and shear while keeping the convergence in microlenses fixed to
$\kappa_{\ast}$=0.09. Each centroid shift is represented against the
magnification in the corresponding axis, $|\mu_{-}|$ and $|\mu_{+}|$
respectively.

###  3.1 Dependence of the centroid shift with the linear macromagnification

We study the centroid displacement of a source of $r_{s}=1R_{E}$ caused by a
population of stars with convergence $\kappa_{\ast}=0.09$ in different
macromagnifications environments. This convergence in microlenses corresponds
to $\alpha=0.2$ for a typical convergence value of $\kappa=0.45$, which is the
average stellar fraction in lens galaxies (Jiménez-Vicente et al., 2015). We
start from the typical values $\kappa=\gamma=0.45$ ($\mu=+10$) for positive
magnification and $\kappa=\gamma=0.55$ ($\mu=-10$) for negative magnification.
Then, in each case we vary either $\kappa$ (without changing $\kappa_{\ast}$)
or $\gamma$ to explore the magnification interval from $\pm 5$ to $\pm 80$.

In Figure 1, we represent the dependence of the centroid shifts, $\Delta
C=\sqrt{\Delta C_{1}^{2}+\Delta C_{2}^{2}}$, with the linear
macromagnification defined as:

$\mu_{\text{lin}}\coloneq\sqrt{\mu_{-}^{2}+\mu_{+}^{2}}=\sqrt{\frac{1}{(1-\kappa-\gamma)^{2}}+\frac{1}{(1-\kappa+\gamma)^{2}}}.$
(3)

According to this trend, the larger the linear macromagnification, the larger
the centroid shifts, as expected, regardless of whether $\kappa$ or $\gamma$
is varied or whether positive or negative magnification is considered. This is
due to the fact that the centroid shift is the square root of the quadratic
sum of the shift in each axis, which in turn can be expressed as the shift
produced only by the stars, $\Delta\vec{C}_{0}$, multiplied by the
magnification of each principal axis:

$\Delta\vec{C}=\left(\begin{array}[]{cc}\frac{1}{1-\kappa-\gamma}&0\\\
0&\frac{1}{1-\kappa+\gamma}\\\ \end{array}\right)\Delta\vec{C}_{0}.$ (4)

In absence of shear, the shifts produced by the stars in each axis are
symmetric (Katz et al., 1986) and we can express the centroid shift as the
displacement in absence of magnification multiplied by the linear
macromagnification:

$\Delta C=\sqrt{\frac{\Delta C_{0,1}^{2}}{(1-\kappa-\gamma)^{2}}+\frac{\Delta
C_{0,2}^{2}}{(1-\kappa+\gamma)^{2}}}\simeq\frac{1}{\sqrt{2}}\,\Delta
C_{0}\,\mu_{\text{lin}},$ (5)

with $\Delta C_{0}=\sqrt{\Delta C_{0,1}^{2}+\Delta
C_{0,2}^{2}}\simeq\sqrt{2}\,|\Delta C_{0,1}|$ as $|\Delta
C_{0,1}|\simeq|\Delta C_{0,2}|$. If the displacement produced by the stars is
constant, then a linear relation is established between $\Delta C$ and
$\mu_{\text{lin}}$.

However, we can see a decrease in the slope of Figure 1 as $\mu_{\text{lin}}$
grows larger, even though the convergence in stars is kept fixed. To explore
this behavior, we show in Figure 2 the contribution to the centroid shifts
from each axis against the corresponding magnification, $\mu_{-}$ and
$\mu_{+}$ in absolute value. The total centroid shift is dominated by the
contribution from axis 1 which has substantially larger magnifications than
axis 2. For the latter direction, the covered range of magnifications is so
small that we cannot consistently recover the linear relation. The expected
increasing trend for the cases ($\mu>0$, $\gamma$=0.45) and ($\mu<0$,
$\kappa$=0.55) is broken for larger $|\mu_{+}|$ which correspond to larger
$\mu_{\text{lin}}$ and the deviation from linearity for the cases ($\mu>0$,
$\kappa$=0.45) and ($\mu<0$, $\gamma$=0.55) occur at smaller $|\mu_{+}|$ which
also belong to the cases of large $\mu_{\text{lin}}$. The decrease in the
slope of $\Delta C_{1}$ is also present at around $|\mu_{-}|\sim 60$. This may
be related to an increase in the number of stars significantly contributing to
the microlensing of the source (departure from the low optical depth
approximation). The number of microlenses in an area of the image plane,
$A_{x}$, back-mapped to an area $A_{y}$ in the source plane is given by

$N_{\ast}=\frac{\kappa_{\ast}}{\pi}|\mu|A_{y}.$ (6)

If we consider that $A_{y}$ subtends the area corresponding to a circular
source size, we obtain $N_{\ast}=\kappa_{\ast}|\mu|r_{s}^{2}$. For the typical
case, $\kappa_{*}=0.09$, $\mu=10$ and $r_{s}=1R_{E}$, there are $N_{\ast}=0.9$
stars in the area of the source. According to Figure 1, the departure from
linearity is notorious when, on average, $\sim 5$ stars in the source area
($|\mu|\sim 60$) are present.

Figure 3: On the left panel, the centroid shift dependence is shown for
different source sizes from 1$R_{E}$ to 90$R_{E}$ with respect to the largest
source of 100$R_{E}$ and for different convergence in microlenses
$\kappa_{\ast}$ from 0.045 to 0.45. On the right panel, the Spearman
correlation coefficient $r$ between magnification and shift for the
$r_{s}$-$\kappa_{\ast}$ pairs considered is displayed. A positive (negative)
correlation depicted in red (in blue) and the complete lack of correlation,
$0$, is presented in white.

In the linear regime, we can extrapolate the centroid shifts in each axis in
absence of macrolensing, $\Delta C_{0,1}$ and $\Delta C_{0,2}$. In order to do
so, we consider two points that have the smallest $\mu_{\text{lin}}$ and,
hence, are least subjected to saturation. By dividing their centroid shifts by
the corresponding magnification and averaging both results, we obtain $\Delta
C_{0,1}=0.681\pm 0.011R_{E}$ and $\Delta C_{0,2}=0.698\pm 0.011R_{E}$ which
are rather similar. We can compare this centroid shift with the probability of
the deflection angle for a distribution of stars, which in the regime of
multiple scatterings the probability distribution function, PDF, is a Gaussian
of width proportional to $\sqrt{\kappa_{\ast}}\theta_{E}$ (Katz et al., 1986;
Petters et al., 2009). Following Equation (17) of Katz et al. (1986), the PDF
width is 1.16$R_{E}$ (for the $N=7.1\cdot 10^{6}$ microlenses considered and
rescaling it to the case of no magnification) which is the same order as the
centroid shift $\Delta C_{0}=0.976\pm 0.011R_{E}$. This value represents the
expected value of the strength of the centroid shift of a source of 1$R_{E}$
for $\kappa_{\ast}=0.09$ when $\kappa=\gamma=0$ ($\mu=1$) and to obtain a
quick estimation for any value of $\kappa$ and $\gamma$ within the linear
regime, we only need to multiply $\Delta C_{0}$ by $\mu_{\text{lin}}/\sqrt{2}$
following the Equation (5).

###  3.2 Dependence of the centroid shift with the source size and the
abundance of microlenses

This is the key study to support the single-epoch method to measure
astrometric microlensing. We will consider several source sizes in Einstein
radius units (from $r_{s}=1$ to $100$ $R_{E}$) and values of $\alpha$ (the
fraction of mass in microlenses, $\alpha=\kappa_{\ast}/\kappa$) ranging from
$\alpha=0.1$ to $1$. We consider a typical case of magnification $\mu=10$ with
$\kappa=\gamma=0.45$. To analyze the combined effect of $\kappa_{\ast}$ and
$r_{s}$ on the centroid shift, we present on the left panel of Figure 3 a 2D
map of the centroid shift as a function of the source size and the fraction of
mass in microlenses, $\Delta C(r_{s},\kappa_{\ast})$. The centroid shifts are
computed with respect to the largest source, in this case $r_{s}=100R_{E}$, to
check whether a large source can be used as a reference for astrometric
microlensing studies.

Figure 4: Deformation of circular sources of radius 1, 5, 10, 50 and 100
$R_{E}$ due to a convergence in microlenses of 0.045 (left) and 0.45 (right).
The scale of the source size radius is shown as a horizontal line on the left.

From Figure 3 it is clear that the centroid shifts decrease with source size
and increase with stellar density. For lower values of convergence in
microlenses, the slope of the decrease is flatter than for large
$\kappa_{\ast}$. However, larger convergences seem to require a larger source
as reference to have a centroid shift close to zero. The average image size
profile of an extended source that is microlensed is a convolution of the
source size with the deviations by the microlenses, i.e.,
$\sqrt{\sigma_{W}^{2}+\kappa_{\ast}\theta_{E}^{2}}$ where $\sigma_{W}$ is the
source width (see, e.g., Katz et al., 1986; Neindorf, 2003; Dai & Pascale,
2021). The profile broadening due to the source size is always symmetrical
(elliptical, in fact, due to the convergence and shear) and does not
contribute to the centroid shift, whereas the second term is related to the
deflection of the stars by the $\kappa_{\ast}$ parameter and can produce
asymmetrical profiles. From this relation, it is clear that the centroid shift
will depend on both the source size and the convergence in stars. For larger
$\kappa_{\ast}$ the profile can be more asymmetrical and the centroid shift
will be larger and, conversely, larger sources will be imaged as large
ellipses that might counteract the shift produced by the microlenses. To grasp
this effect, we show in Figure 4 the images of several sources for the
smallest and largest $\kappa_{\ast}$ considered. The stellar distribution
chosen among the 1000 realizations is the one which has shifts close to the
average for the smallest source. For a source of 1$R_{E}$, the image is quite
distorted but as the size grows larger, the images become more regular as if
all the mass were distributed smoothly. For the case of largest
$\kappa_{\ast}$, the images are more distorted than for smaller
$\kappa_{\ast}$ due to the fact that there are more stars in the vicinity of
the source that affect the light rays and leads to larger shifts.

###  3.3 Correlation between the centroid shifts and the flux magnification
induced by microlensing

Figure 5: Scatter plots of centroid shifts, $\Delta C$, and magnifications,
$\mu$, for the typical case $\alpha=0.2$ ($\kappa_{\ast}=0.09$). The source
size is indicated in the upper left corner of each plot and the Spearman r
coefficient is in the upper right corner.

Since the microlensing-induced shifts are expected to be small, one possible
strategy for detecting them is to observe the lens systems at epochs when
large shifts are more likely to occur. To detect possible correlations for our
set of simulations, we compute for every realization of the microlens
distribution of the previous subsection their corresponding magnification map
with the FMM-IPM code (Jiménez-Vicente & Mediavilla, 2022) and convolve them
with the source profiles. For each pair of $r_{s}$ and $\kappa_{\ast}$ we
compute the Spearman correlation coefficient $r$ between the magnification at
the source position and the centroid shift of their corresponding 1000
realizations. We choose the Spearman correlation coefficient over the Pearson
correlation coefficient because the former does not assume a linear
relationship between parameters, only a monotonically increasing or decreasing
relationship. On the right panel of Figure 3 we show the correlation
coefficient in a 2D map of $\kappa_{\ast}$ versus $r_{s}$. The average
correlation and its standard deviation only reaches the $-0.005\pm 0.016$
level, and the maximum and minimum correlation coefficients are $+0.086$ and
$-0.084$, respectively. We can argue, hence, that there is no correlation
between magnification and centroid shift in random realizations of the
microlens distribution. One interpretation of this finding might be that
images with a large number of microimages (high magnification) are not
guaranteed to be asymmetrically distributed (large shift). Also the opposite
is not guaranteed, that is, low magnified images may not have their
microimages symmetrically spread.

As an illustrative example of the scatter between centroid shifts and
magnifications, Figure 5 shows the distribution of these parameters for all
sizes considered for the typical microlens mass fraction $\alpha=0.2$
($\kappa_{\ast}=0.09$). There is no obvious correlation between the parameters
but it must be noted that for sizes above 40$R_{E}$ the variance does not
exceed more than (9.5, 10.5) in magnification. In such cases, the possible
correlation between shift and magnification might be diluted and such effect
must be taken into account. In our case, however, the lack of correlation
obtained for smaller sizes (with the same order of magnitude as for larger
sources) points out that the centroid shifts obtained from different random
stellar distributions are not correlated with the magnifications irrespective
of the source size.

Given the lack of correlation, the choice of high-magnification events in
single-epoch measurements does not guarantee an increase of centroid shifts
because of the random distributions of microlenses. Lewis & Ibata (1998) found
a slight correlation between these parameters when the source was placed at
subsequent positions of the magnification map but no correlation coefficient
was reported. Treyer & Wambsganss (2004) found positive correlations when the
source is moving in the magnification map and specially strong correlations
are found when the source traverses its own size and crosses a caustic. Hence,
in differential astrometric campaigns where the position of the image is
tracked during a certain period of time, large magnification events may be
used as alerts to intensively measure the image position as its shift is
likely to be greater. However, in single-epoch measurements, the shift with
respect to an unmicrolensed reference it is not guaranteed to be larger than
in low magnification events.

Figure 6: Occurrence rate of centroid shifts for the four simulated lens
systems. The upper left panel shows the shifts of a $0.3R_{E}$ source in the
image A of the lens system Q2237+030. The upper right panel shows the shifts
of a $0.06R_{E}$ source in the image A of the lens system RXJ1131$-$1231\. The
lower left panel shows the shifts of a $0.7R_{E}$ source in the image A2 of
the lens system PG1115+080. The lower right panel shows the shifts of a
$0.6R_{E}$ source in the image A of the lens system SDSS J1004+4112.

##  4 Specific study of the best-suited cases

According to the previous simulations, the best-suited systems should be those
with large abundance of microlenses, $\kappa_{\ast}$, and linear
magnification, $\mu_{lin}$, but small source size, $r_{s}$. Also, all these
results were reported in units of the Einstein angle, $\theta_{E}$, so the
physical angular shift will be directly proportional to $\theta_{E}$, which in
turn depends on the redshifts of the source and lens and the mean mass of
microlenses. We can estimate the order of centroid shifts of specific lens
systems by converting the centroid shifts of Figure 3 in each direction in
shifts in absence of macrolensing, $\Delta\vec{C}$. Then, we interpolate from
the 2D maps the centroid shift in absence of macrolensing at the specific
values of ($r_{s}$,$\kappa_{\ast}$) for each lens system. This can be
converted into the estimation of the centroid shift by following Equation 4
and multiplying it by the Einstein radius in $\mu$as. Since we need to know
$\kappa$, $\kappa_{\ast}$, $\gamma$, $r_{s}$ and $\theta_{E}$, we only
considered the lens systems in the GERLUMPH222http://gerlumph.swin.edu.au/
macromodel database (Vernardos et al., 2014). $r_{s}$ and $\theta_{E}$ are
taken from Mosquera & Kochanek (2011), assuming stars with a mean mass
$\langle M\rangle=0.3M_{\odot}$ as microlenses. $\kappa_{\ast}$ corresponds to
$\alpha=\kappa_{\ast}/\kappa=0.2$, which is the typical stellar fraction for
galaxies (Jiménez-Vicente et al., 2015), unless otherwise stated. The centroid
shift estimations of each image from the lens systems considered are reported
in Appendix A. From these estimates, we select the following interesting cases
to accurately simulate their centroid shifts.

###  4.1 Image A of Q2237+030

We model the lens system Q2237+030 because it has the largest centroid shift
prediction in the sample. This gravitational lens was already pointed out by
Lewis & Ibata (1998) and Treyer & Wambsganss (2004) as an ideal case for
astrometric microlensing. We simulate 1000 centroid shift realizations using
$\kappa=0.4$ and $\gamma=0.4$ from the macromodel of Poindexter & Kochanek
(2010). We consider the usual value of $\alpha=1$ for this system given that
the images are located in the inner part of the lens galaxy. The source is a
uniform circle of $0.3R_{E}$ whose half-light radius corresponds to the value
obtained by Muñoz et al. (2016) shifted to the B filter central wavelength
($\lambda_{\text{obs}}=4450$Å) using their inferred wavelength dependence of
$p=0.66$. The histogram of the expected shifts for this system is shown in the
upper left panel of Figure 6. The centroid shifts can reach up to 100 $\mu$as
with a median value of 21 $\mu$as.

###  4.2 Image A of RXJ1131$-$1231

Image A of the lens system RXJ1131$-$1231 is the fourth image with the largest
predicted centroid shift but the second lens system with largest $\Delta C$.
The macromodel parameters for image A are $\kappa=0.569$, $\gamma=0.465$ and
$\alpha=0.11$, extracted from the mass model of Dai et al. (2010) with
$f_{\ast}=0.3$, because this is the mass model favored by the microlensing
study performed in the same work. We place a circular source profile of
$0.06R_{E}$, which corresponds to the size at $\lambda_{\text{obs}}=4450$Å
(Dai et al., 2010), following a thin-disk wavelength dependence, $p=4/3$. In
the upper right panel of Figure 6 half of the cases have shifts above 14
$\mu$as and the maximum shift achieved is around 60 $\mu$as.

###  4.3 Image A2 of PG1115+080

The image A2 of the lens system PG1115+080 is the next image to have large
centroid shifts according to our estimations. We model the centroid shifts by
using the mass model of Morgan et al. (2008b) ($\kappa$=0.61, $\gamma$=0.44
and $\alpha=0.12$) and their inferred optical size from microlensing, which
favors the $f_{\ast}=0.4$ case. The circular face-on size in the center of the
B-band following the thin-disk size-wavelength dependence is $0.7R_{E}$. The
histogram of 1000 centroid shifts from different random stellar distributions
is depicted in the lower left panel of Figure 6. For this system, the centroid
shifts can be as large as 50 $\mu$as but the median is 12 $\mu$as.

###  4.4 Image A of SDSS J1004+4112

Despite that the image A of SDSS J1004+4112 is the 16th image with the largest
$\Delta C$, it is worth studying its centroid shifts given that photometric
microlensing variability has been detected even though a galaxy cluster is
acting as a lens and has a magnification much larger than most typical galaxy
lenses. The centroid shifts are computed for $\kappa=0.73$ and $\gamma=0.33$
from Forés-Toribio et al. (2022). We adopt a stellar fraction of
$\alpha=0.075$ and a circular source size of $0.6R_{E}$ based on the
inferences in Forés-Toribio et al. (2024, submitted) assuming a thin-disk
wavelength dependence to estimate the source size at
$\lambda_{\text{obs}}=4450$ Å. In this case, the centroid shifts reach up to
25 $\mu$as and with a median value of 5 $\mu$as, smaller than the rest, but
compatible with what was expected in Table A.

##  5 Discussion

It is convenient to discuss the possible applications of astrometric
microlensing attending to the amplitude of the centroid shift, which depends
on the mass of the microlens. For typical configurations of lens galaxies and
quasar sources
$|\Delta\vec{C}|\sim\sqrt{\mu_{+}^{2}+\mu_{-}^{2}}\sqrt{M/0.3M_{\odot}}\,\rm\mu
as$. For linear magnifications about 10 and stellar masses of $0.3M_{\odot}$,
we obtain centroid shifts of the order tens of $\mu$as (microlensing scale),
while for masses above $10^{3}M_{\odot}$, we obtain shifts of the order of mas
(millilensing scale). Between these scales, the mass range from $10$ to
$10^{2}$ $M_{\odot}$ has special interest because it corresponds to the masses
of the black holes (BHs) detected by the LIGO–Virgo–KAGRA Collaboration
(Abbott et al., 2023).

It should be noted that the results of the previous sections are computed for
distributions of microlenses of the same mass. However, the actual mass
distributions of stars or other compact objects have a mass spectrum that may
alter the results to some extent. It is commonly accepted that the average
mass of the distribution is the representative quantity and the results can be
scaled to this mass. Katz et al. (1986) (Equation 24b) found that the
deflection by a spectrum mass of microlenses mainly depends on $\langle
m^{2}\rangle/\langle m\rangle$ along with the average $\kappa_{\ast}$. Hence,
when a large range in the mass of the microlenses is considered, the
deflections not only depend on the average mass (see, e.g., Esteban-Gutiérrez
et al., 2020; Dai & Pascale, 2021). For the sake of simplicity, in the
following sections we will only discuss the effect of microlenses of different
mass ranges separately. A joint analysis with all possible microlens masses is
beyond the scope of the paper but it should be performed to properly assess
the impact of each population on the astrometry.

###  5.1 Stellar mass lenses, $M\sim M_{\odot}$ (microlensing): multiply
imaged quasars

Since there is a dependence between the observed wavelength and the size of
the emitting region, the continua at different wavelengths should experiment
centroid shifts of different amplitude according to our simulations. This will
help to study the structure of the accretion disk, in particular its
temperature profile. The same reasoning can be applied to the BLR, which is
commonly accepted to be stratified according to the line ionization degree.

###  5.2 Intermediate-mass lenses, $10M_{\odot}\lesssim M\lesssim
10^{2}M_{\odot}$: LIGO-Virgo-KAGRA BHs

This range of masses is very interesting because there is evidence from LIGO-
Virgo-KAGRA experiments of the existence of BHs with masses in this interval.
Shifts of order $10^{2}$ $\mu$as are expected, however, the Einstein radius of
the BHs would be around $10^{2}$ light-days and, according to Figure 3, a
reference source insensitive to microlensing should have a size of $10^{4}$
light-days, significantly exceeding the size of BLRs. An option to apply the
single-epoch method above this mass might be to adopt as a reference a larger
region (e.g., the infrared torus or the extended radio emission region).
Alternatively, we can measure the differences between the positions of several
images of a gravitationally lensed quasar at different epochs, namely a
differential astrometric microlensing approach. The relative shifts between
images should have a random origin and be, consequently, uncorrelated.

###  5.3 Large-mass lenses, $10^{3}M_{\odot}\lesssim M\lesssim
10^{6}M_{\odot}$ (millilensing): PMBHs, PBH clusters

Shifts in the 0$\farcs$001 to 0$\farcs$01 range are expected in this interval
of masses, which could be routinely measured with normal CCD photometry in a
medium sized telescope. Comparison of astrometry from the literature of HST,
JWST and/or ground based telescopes of quads taken at different epochs should
be useful to put constraints on any population of compact objects in this mass
interval. If a significant population of millilenses of these masses exists,
it should have a strong impact on lens-model fitting, as the induced centroid
shifts exceeds the uncertainties in astrometric models.

###  5.4 Experimental approach

There are several instrumental possibilities, present and future, to perform
astrometric measurements with precisions similar to those required by our
calculations. With the ESA’s Gaia space mission, high-precision differential
astrometry in the separation of relatively bright lensed quasar images with
accuracies of a few $\mu$as may be obtained in the optical band (Gaia
Collaboration et al., 2016). This precision can be improved with GRAVITY,
which combines the near-infrared light of the four 8.2m telescope of ESO’s
VLT-I (GRAVITY Collaboration et al., 2021), or with Theia, which promise
submicroarcsecond astrometric precisions. The next generation of instruments
like HARMONI, designed to obtain integral field spectroscopy at the maximum
spatial resolution of the E-ELT, may be used to apply the single-epoch method
to a large number of quasar images.

##  6 Conclusions

The main conclusions that can be derived from our simulations are the
following:

1\. The amplitude of the centroid shifts, $\Delta C$, increases with the
linear magnification, $\mu_{\text{lin}}=\sqrt{\mu_{-}^{2}+\mu_{+}^{2}}$,
almost linearly. Hence, the amplitude of the centroid shifts can be estimated
as the product of the centroid shift in absence of macromagnification,
$|\Delta\vec{C}_{0}|$, and the linear macromagnification (see Equation 5).
Theoretical analyses of the random deflections of microlenses (Katz et al.,
1986; Petters et al., 2009) show that the amplitude depends on the square root
of the convergence in microlenses and is directly proportional to the Einstein
angle $\theta_{E}$. The $|\Delta\vec{C}_{0}|$ found in our simulations is
similar to the expected PDF width of the deflection from Katz et al. (1986).

2\. The average amplitude of the centroid shift increases with the convergence
of microlenses and decreases with size with a gentle slope, approximating zero
only for size values of around 100 Einstein radii. The dependence of the
average image profile on these two parameters has been already proven (Dai &
Pascale, 2021) but in Figure 3 the impact on the centroid shift has been
computed numerically. The gentle size dependence limits the use of the BLR as
reference for zero centroid shift for microlenses masses $M\lesssim
10M_{\odot}$. Above this mass, the BLR becomes sensitive to astrometric333We
notice that in the case of flux magnification induced by microlensing the BLR
becomes critically sensitive to microlensing when the Einstein radius of the
microlenses is comparable to the size of the region, and this typically occurs
for $M\gtrsim 100M_{\odot}$. microlensing and we need larger emission regions
(e.g., the infrared torus or the extended radio emission region) as reference
to apply the single-epoch method or, alternatively, to consider differential
measurements of the distances between images in a gravitationally lensed
quasar in several epochs.

3\. We have found no correlation between centroid shift and magnification
(macromagnification plus micromagnification) between random distributions of
microlenses. Hence, a single-epoch astrometric observation during a high-
magnification event does not guarantee finding large centroid shift with
respect to the baseline. However, in the differential astrometry approach, a
correlation arises (Lewis & Ibata, 1998; Treyer & Wambsganss, 2004) due to the
tracking during subsequent source positions.

4\. We select four gravitational lensed images of quasars which will be prone
to astrometric microlensing and study the centroid displacements. The image A
of Q2237+030 is the most favorable system with a 50% chance of measuring
centroid shifts larger than 21 $\mu$as for a source size of 0.3$R_{E}$ and
microlens masses of 0.3$M_{\odot}$.

5\. Strategy for measurements: We propose to use spectro-astrometry in
upcoming integral field units to simultaneously obtain single-epoch
photocenters of the continuum and of different emission line regions when
probing microlenses of stellar masses and stratification of quasar accretion
disks. If a large emission region for reference is not available, then a
differential astrometry approach can be applied between multiple images of
lensed quasars in different epochs with high-precision instrumentation.

We thank the referee for their comments and suggestions, which have
significantly improved this work. This research was supported by the grants
PID2020-118687GB-C31, PID2020-118687GB-C32 and PID2020-118687GB-C33 financed
by the Spanish Ministerio de Ciencia e Innovación through
MCIN/AEI/10.13039/501100011033. J.A.M. is also supported by the Generalitat
Valenciana with the project of excellence Prometeo/2020/085. J.J.V. is also
supported by the project FQM-108, financed by Junta de Andalucía. CdB
acknowledges support from a Beatriz Galindo senior fellowship (BG22/00166)
from the Spanish Ministry of Science, Innovation and Universities.

##  Appendix A Estimated centroid shifts for specific lens systems

The expected centroid shifts following the procedure described in §4 are
presented in Table A in decreasing order of $\Delta C$. The sources radii in
Mosquera & Kochanek (2011) are estimated using their I-band magnitudes and a
thin-disk model (Shakura & Sunyaev, 1973). This method typically
underestimates the source sizes in comparison to estimations from black hole
masses and luminosities using the same thin-disk theory as stated in Mosquera
& Kochanek (2011). To obtain a safe estimate of the centroid shifts, we
rescale the quasar sizes to a median value of $5$ light-days as the typical
size obtained with the latter scaling relation (see e.g. Yu et al. (2020) for
a black hole mass of $10^{9}M_{\sun}$ and an Eddington ratio of
$\dot{m}_{E}=0.3$).

Table 1: Estimation of centroid shifts, $\Delta C$, for 79 lensed quasar
images in the GERLUMPH database (Vernardos et al., 2014).

Lens system | Image | $\kappa$ | $\gamma$ | $\alpha$ |  $R_{1/2}$ ($\theta_{E}$) |  $\theta_{E}$ ($\mu$as) |  $\Delta C$ ($\mu$as) | References
---|---|---|---|---|---|---|---|---
Q2237+030 | A | 0.4 | 0.4 | 1.0 | 1.05 | 3.90 | 21.17 | (5)  |  |  |
Q2237+030 | D | 0.62 | 0.62 | 1.0 | 1.05 | 3.90 |  $\gtrsim$19.24 (14.15)aaLower level estimations given that their $\kappa_{\ast}$ are larger than the maximum $\kappa_{\ast}$ in Figure 3. The result using the PDF width of random microlensing deflections of Katz et al. (1986) with $N=10^{6}$ microlenses as an estimate of $\Delta C_{0}$ is reported in parentheses. | (5)  |  |  |
Q2237+030 | B | 0.38 | 0.39 | 1.0 | 1.05 | 3.90 | 17.93 | (5)  |  |  |
RXJ1131$-$1231 | A | 0.569 | 0.465 | 0.11 | 0.94 | 1.19 | 14.01 | (4)  |  |  |
PG1115+080 | A2 | 0.61 | 0.44 | 0.12 | 1.20 | 1.43 | 12.00 | (2)  |  |  |
PG1115+080 | A1 | 0.59 | 0.36 | 0.11 | 1.20 | 1.43 | 11.17 | (2)  |  |  |
Q2237+030 | C | 0.73 | 0.72 | 1.0 | 1.05 | 3.90 |  $\gtrsim$10.84 (6.22)aaLower level estimations given that their $\kappa_{\ast}$ are larger than the maximum $\kappa_{\ast}$ in Figure 3. The result using the PDF width of random microlensing deflections of Katz et al. (1986) with $N=10^{6}$ microlenses as an estimate of $\Delta C_{0}$ is reported in parentheses. | (5)  |  |  |
HE0047$-$1756 | A | 0.45 | 0.48 | 0.2 | 1.47 | 1.23 | 8.15 | (8)  |  |  |
MG0414+0534 | A2 | 0.53 | 0.524 | 0.2 | 0.34 | 0.85 | 8.01 | (7)  |  |  |
SDSSJ0924+0219 | A | 0.39 | 0.54 | 0.2 | 0.71 | 1.24 | 7.90 | (9)  |  |  |
MG0414+0534 | A1 | 0.489 | 0.454 | 0.2 | 0.34 | 0.85 | 7.36 | (7)  |  |  |
RXJ1131$-$1231 | B | 0.53 | 0.41 | 0.099 | 0.94 | 1.19 | 7.29 | (4)  |  |  |
SDSSJ0924+0219 | D | 0.4 | 0.68 | 0.2 | 0.71 | 1.24 | 6.99 | (9)  |  |  |
RXJ1131$-$1231 | C | 0.546 | 0.387 | 0.103 | 0.94 | 1.19 | 6.77 | (4)  |  |  |
HE0230$-$2130 | B | 0.51 | 0.587 | 0.2 | 0.72 | 1.12 | 5.77 | (7)  |  |  |
SDSSJ1004+4112 | A | 0.73 | 0.33 | 0.075 | 1.07 | 0.93 | 5.65 | (10)  |  |  |
B1422+231 | B | 0.492 | 0.628 | 0.2 | 2.69 | 1.44 | 5.59 | (7)  |  |  |
HE0230$-$2130 | A | 0.472 | 0.416 | 0.2 | 0.72 | 1.12 | 4.87 | (7)  |  |  |
SDSSJ1138+0314 | D | 0.523 | 0.614 | 0.2 | 0.54 | 1.23 | 4.57 | (7)  |  |  |
SDSSJ0924+0219 | C | 0.44 | 0.7 | 0.24 | 0.71 | 1.24 | 4.53 | (9)  |  |  |
HE0435$-$1223 | B | 0.539 | 0.602 | 0.2 | 0.95 | 1.16 | 4.20 | (7)  |  |  |
SDSSJ1138+0314 | A | 0.465 | 0.384 | 0.2 | 0.54 | 1.23 | 3.99 | (7)  |  |  |
RXJ0911+0551 | B | 0.586 | 0.281 | 0.2 | 0.94 | 0.97 | 3.91 | (7)  |  |  |
B1422+231 | A | 0.38 | 0.473 | 0.2 | 2.69 | 1.44 | 3.90 | (7)  |  |  |
SDSSJ1206+4332 | A | 0.43 | 0.41 | 0.2 | 0.84 | 1.23 | 3.60 | (3)  |  |  |
HE0435$-$1223 | C | 0.444 | 0.396 | 0.2 | 0.95 | 1.16 | 3.44 | (7)  |  |  |
QJ0158$-$4325 | A | 0.54 | 0.28 | 0.15 | 1.74 | 1.36 | 3.32 | (1)  |  |  |
HE0435$-$1223 | A | 0.445 | 0.383 | 0.2 | 0.95 | 1.16 | 3.21 | (7)  |  |  |
SDSSJ0924+0219 | B | 0.38 | 0.45 | 0.19 | 0.71 | 1.24 | 3.17 | (9)  |  |  |
HE1104$-$1805 | A | 0.64 | 0.52 | 0.2 | 3.77 | 0.96 | 3.15 | (3)  |  |  |
QJ0158$-$4325 | B | 0.89 | 0.57 | 0.32 | 1.74 | 1.36 | 3.10 | (1)  |  |  |
RXJ0911+0551 | A | 0.646 | 0.544 | 0.2 | 0.94 | 0.97 | 2.87 | (7)  |  |  |
SDSSJ1138+0314 | C | 0.438 | 0.349 | 0.2 | 0.54 | 1.23 | 2.79 | (7)  |  |  |
PG1115+080 | B | 0.69 | 0.58 | 0.15 | 1.20 | 1.43 | 2.73 | (2)  |  |  |
SDSSJ1138+0314 | B | 0.578 | 0.673 | 0.2 | 0.54 | 1.23 | 2.66 | (7)  |  |  |
HE0435$-$1223 | D | 0.587 | 0.648 | 0.2 | 0.95 | 1.16 | 2.66 | (7)  |  |  |
RXJ0911+0551 | C | 0.637 | 0.577 | 0.2 | 0.94 | 0.97 | 2.54 | (7)  |  |  |
H1413+117 | A | 0.53 | 0.64 | 0.2 | 1.86 | 0.86 | 2.51 | (3)  |  |  |
HE0512$-$3329 | A | 0.59 | 0.55 | 0.2 | 3.76 | 0.69 | 2.47 | (3)  |  |  |
HE0047$-$1756 | B | 0.62 | 0.66 | 0.2 | 1.47 | 1.23 | 2.41 | (8)  |  |  |
HE0230$-$2130 | C | 0.44 | 0.334 | 0.2 | 0.72 | 1.12 | 2.40 | (7)  |  |  |
SDSSJ1155+6346 | B | 1.67 | 1.47 | 0.2 | 1.53 | 1.92 | 2.28 | (8)  |  |  |
B1422+231 | C | 0.365 | 0.378 | 0.2 | 2.69 | 1.44 | 2.22 | (7)  |  |  |
PG1115+080 | C | 0.54 | 0.25 | 0.08 | 1.20 | 1.43 | 2.19 | (2)  |  |  |
RXJ1131$-$1231 | D | 1.001 | 0.635 | 0.242 | 0.94 | 1.19 | 2.19 | (4)  |  |  |
H1413+117 | C | 0.46 | 0.35 | 0.2 | 1.86 | 0.86 | 2.13 | (3)  |  |  |
WFI2033$-$4723 | A1 | 0.506 | 0.255 | 0.2 | 1.10 | 0.93 | 2.02 | (7)  |  |  |
SDSSJ1206+4332 | B | 0.63 | 0.72 | 0.2 | 0.84 | 1.23 | 2.02 | (3)  |  |  |
MG0414+0534 | B | 0.46 | 0.316 | 0.2 | 0.34 | 0.85 | 1.89 | (7)  |  |  |
SDSSJ1004+4112 | B | 0.65 | 0.23 | 0.048 | 1.07 | 0.93 | 1.86 | (10)  |  |  |
Q1017$-$207 | A | 0.35 | 0.45 | 0.2 | 3.21 | 0.95 | 1.83 | (3)  |  |  |
WFI2033$-$4723 | A2 | 0.665 | 0.643 | 0.2 | 1.10 | 0.93 | 1.76 | (7)  |  |  |
SDSSJ1001+5027 | B | 0.74 | 0.72 | 0.2 | 1.91 | 1.22 | 1.71 | (3)  |  |  |
H1413+117 | B | 0.43 | 0.34 | 0.2 | 1.86 | 0.86 | 1.71 | (3)  |  |  |
H1413+117 | D | 0.58 | 0.69 | 0.2 | 1.86 | 0.86 | 1.67 | (3)  |  |  |
SDSSJ1353+1138 | B | 0.96 | 0.89 | 0.2 | 2.80 | 1.44 | 1.53 | (3)  |  |  |
FBQ0951+2635 | B | 1.07 | 1.02 | 0.2 | 3.67 | 1.57 | 1.49 | (3)  |  |  |
HE0230$-$2130 | D | 1.07 | 0.864 | 0.2 | 0.72 | 1.12 | 1.44 | (7)  |  |  |
WFI2033$-$4723 | B | 0.392 | 0.302 | 0.2 | 1.10 | 0.93 | 1.41 | (7)  |  |  |
MG0414+0534 | C | 0.676 | 0.693 | 0.2 | 0.34 | 0.85 | 1.40 | (7)  |  |  |
SDSSJ1001+5027 | A | 0.35 | 0.28 | 0.2 | 1.91 | 1.22 | 1.37 | (3)  |  |  |
WFI2033$-$4723 | C | 0.7 | 0.735 | 0.2 | 1.10 | 0.93 | 1.34 | (7)  |  |  |
Q0957+561 | B | 1.03 | 0.91 | 0.2 | 3.51 | 1.28 | 1.34 | (3)  |  |  |
SDSSJ0806+2006 | B | 0.82 | 0.77 | 0.2 | 1.16 | 1.04 | 1.33 | (3)  |  |  |
HE0512$-$3329 | B | 0.41 | 0.37 | 0.2 | 3.76 | 0.69 | 1.32 | (3)  |  |  |
SDSSJ0806+2006 | A | 0.34 | 0.29 | 0.2 | 1.16 | 1.04 | 1.22 | (3)  |  |  |
SDSSJ1353+1138 | A | 0.3 | 0.22 | 0.2 | 2.80 | 1.44 | 1.19 | (3)  |  |  |
FBQ0951+2635 | A | 0.28 | 0.15 | 0.2 | 3.67 | 1.57 | 1.12 | (3)  |  |  |
SDSSJ1004+4112 | C | 0.59 | 0.23 |  $<$0.045 | 1.07 | 0.93 |  $<$1.09 | (10)  |  |  |
SDSSJ1155+6346 | A | 0.22 | 0.03 | 0.2 | 1.53 | 1.92 | 1.09 | (8)  |  |  |
SDSSJ1004+4112 | D | 1.02 | 0.49 | 0.066 | 1.07 | 0.93 | 1.08 | (10)  |  |  |
SBS1520+530 | B | 0.9 | 0.85 | 0.2 | 2.80 | 0.91 | 1.00 | (3)  |  |  |
SBS0909+523 | A | 0.66 | 0.77 | 0.2 | 9.12 | 0.71 | 0.88 | (6)  |  |  |
HE1104$-$1805 | B | 0.33 | 0.21 | 0.2 | 3.77 | 0.96 | 0.86 | (3)  |  |  |
Q1017$-$207 | B | 1.23 | 1.32 | 0.2 | 3.21 | 0.95 | 0.79 | (3)  |  |  |
RXJ0911+0551 | D | 0.29 | 0.066 | 0.2 | 0.94 | 0.97 | 0.74 | (7)  |  |  |
SBS1520+530 | A | 0.29 | 0.15 | 0.2 | 2.80 | 0.91 | 0.68 | (3)  |  |  |
Q0957+561 | A | 0.2 | 0.15 | 0.2 | 3.51 | 1.28 | 0.67 | (3)  |  |  |
SBS0909+523 | B | 0.36 | 0.25 | 0.2 | 9.12 | 0.71 | 0.67 | (6)  |  |  |

Note. — The macromodels used ($\kappa$, $\gamma$ and $\alpha$) are shown along
with their references. The half-light radius, $R_{1/2}$, and the Einstein
angle, $\theta_{E}$, are taken from Mosquera & Kochanek (2011).

References. — (1) $\kappa$, $\gamma$ and $\alpha$ from Morgan et al. (2008a),
$f_{M/L}=0.4$ mass model ($\alpha$ closest to 0.2). (2) $\kappa$, $\gamma$ and
$\alpha$ from Morgan et al. (2008b), $f_{M/L}=0.4$ mass model favored by
microlensing. (3) Mediavilla et al. (2009), (4) $\kappa$, $\gamma$ and
$\alpha$ from Dai et al. (2010), $f_{M/L}=0.3$ mass model favored by
microlensing. (5) $\kappa$ and $\gamma$ from Poindexter & Kochanek (2010) and
$\alpha$ from, e.g., Muñoz et al. (2016). (6) Mediavilla et al. (2011). (7)
Schechter et al. (2014). (8) Rojas et al. (2014). (9) $\kappa$, $\gamma$ and
$\alpha$ from MacLeod et al. (2015), $f_{M/L}=0.6$ mass model ($\alpha$
closest to 0.2). (10) $\kappa$ and $\gamma$ from Forés-Toribio et al. (2022)
and $\alpha$ from Forés-Toribio et al. (2024, submitted).

## References

  * Abbott et al. (2023) Abbott, R., Abbott, T. D., Acernese, F., et al. 2023, Physical Review X, 13, 041039
  * Awad et al. (2023) Awad, P., Chan, J. H. H., Millon, M., Courbin, F., & Paic, E. 2023, A&A, 673, A88
  * Chang & Refsdal (1979) Chang, K., & Refsdal, S. 1979, Nature, 282, 561
  * Chen et al. (2007) Chen, J., Rozo, E., Dalal, N., & Taylor, J. E. 2007, ApJ, 659, 52
  * Dai & Pascale (2021) Dai, L., & Pascale, M. 2021, arXiv e-prints, arXiv:2104.12009
  * Dai et al. (2010) Dai, X., Kochanek, C. S., Chartas, G., et al. 2010, ApJ, 709, 278
  * Eigenbrod et al. (2008) Eigenbrod, A., Courbin, F., Meylan, G., et al. 2008, A&A, 490, 933
  * Esteban-Gutiérrez et al. (2020) Esteban-Gutiérrez, A., Agües-Paszkowsky, N., Mediavilla, E., et al. 2020, ApJ, 904, 176
  * Fian et al. (2023) Fian, C., Muñoz, J. A., Mediavilla, E., et al. 2023, A&A, 678, A108
  * Forés-Toribio et al. (2024) Forés-Toribio, R., Muñoz, J. A., Fian, C., Jiménez-Vicente, J., & Mediavilla, E. 2024, A&A, submitted
  * Forés-Toribio et al. (2022) Forés-Toribio, R., Muñoz, J. A., Kochanek, C. S., & Mediavilla, E. 2022, ApJ, 937, 35
  * Gaia Collaboration et al. (2016) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1
  * GRAVITY Collaboration et al. (2021) GRAVITY Collaboration, Abuter, R., Amorim, A., et al. 2021, A&A, 647, A59
  * Honma & Kurayama (2002) Honma, M., & Kurayama, T. 2002, ApJ, 568, 717
  * Jiménez-Vicente & Mediavilla (2019) Jiménez-Vicente, J., & Mediavilla, E. 2019, ApJ, 885, 75
  * Jiménez-Vicente & Mediavilla (2022) —. 2022, ApJ, 941, 80
  * Jiménez-Vicente et al. (2015) Jiménez-Vicente, J., Mediavilla, E., Kochanek, C. S., & Muñoz, J. A. 2015, ApJ, 799, 149
  * Jiménez-Vicente et al. (2014) Jiménez-Vicente, J., Mediavilla, E., Kochanek, C. S., et al. 2014, ApJ, 783, 47
  * Katz et al. (1986) Katz, N., Balbus, S., & Paczynski, B. 1986, ApJ, 306, 2
  * Kochanek (2004) Kochanek, C. S. 2004, ApJ, 605, 58
  * Lewis & Ibata (1998) Lewis, G. F., & Ibata, R. A. 1998, ApJ, 501, 478
  * MacLeod et al. (2015) MacLeod, C. L., Morgan, C. W., Mosquera, A., et al. 2015, ApJ, 806, 258
  * Mediavilla et al. (2009) Mediavilla, E., Muñoz, J. A., Falco, E., et al. 2009, ApJ, 706, 1451
  * Mediavilla et al. (2011) Mediavilla, E., Muñoz, J. A., Kochanek, C. S., et al. 2011, ApJ, 730, 16
  * Morgan et al. (2008a) Morgan, C. W., Eyler, M. E., Kochanek, C. S., et al. 2008a, ApJ, 676, 80
  * Morgan et al. (2008b) Morgan, C. W., Kochanek, C. S., Dai, X., Morgan, N. D., & Falco, E. E. 2008b, ApJ, 689, 755
  * Morgan et al. (2012) Morgan, C. W., Hainline, L. J., Chen, B., et al. 2012, ApJ, 756, 52
  * Mosquera & Kochanek (2011) Mosquera, A. M., & Kochanek, C. S. 2011, ApJ, 738, 96
  * Motta et al. (2012) Motta, V., Mediavilla, E., Falco, E., & Muñoz, J. A. 2012, ApJ, 755, 82
  * Muñoz et al. (2016) Muñoz, J. A., Vives-Arias, H., Mosquera, A. M., et al. 2016, ApJ, 817, 155
  * Neindorf (2003) Neindorf, B. 2003, A&A, 404, 83
  * Petters et al. (2009) Petters, A. O., Rider, B., & Teguia, A. M. 2009, Journal of Mathematical Physics, 50, 072503
  * Poindexter & Kochanek (2010) Poindexter, S., & Kochanek, C. S. 2010, ApJ, 712, 658
  * Popović & Simić (2013) Popović, L. Č., & Simić, S. 2013, MNRAS, 432, 848
  * Rojas et al. (2014) Rojas, K., Motta, V., Mediavilla, E., et al. 2014, ApJ, 797, 61
  * Salata & Zhdanov (2003) Salata, S. A., & Zhdanov, V. I. 2003, AJ, 125, 1033
  * Schechter et al. (2014) Schechter, P. L., Pooley, D., Blackburne, J. A., & Wambsganss, J. 2014, ApJ, 793, 96
  * Shakura & Sunyaev (1973) Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337
  * Sluse et al. (2012) Sluse, D., Chantry, V., Magain, P., Courbin, F., & Meylan, G. 2012, A&A, 538, A99
  * Treyer & Wambsganss (2004) Treyer, M., & Wambsganss, J. 2004, A&A, 416, 19
  * Vernardos et al. (2014) Vernardos, G., Fluke, C. J., Bate, N. F., & Croton, D. 2014, ApJS, 211, 16
  * Wambsganss (2006) Wambsganss, J. 2006, in Saas-Fee Advanced Course 33: Gravitational Lensing: Strong, Weak and Micro, ed. G. Meylan, P. Jetzer, P. North, P. Schneider, C. S. Kochanek, & J. Wambsganss, 453–540
  * Yu et al. (2020) Yu, Z., Martini, P., Davis, T. M., et al. 2020, ApJS, 246, 16
  * Zheng et al. (2022) Zheng, W., Chen, X., Li, G., & Chen, H.-Z. 2022, ApJ, 931, 114