The wave at the plane of the photodetector will be, according to eq. C.7:

$E_{k}(r,\phi)\propto\int_{0}^{\frac{w}{2}}\text{d}\rho\int_{0}^{2\pi}\text{d}\theta\,\rho\exp\left[ik\left(\frac{\rho^{2}}{2}\left(\frac{1}{z_{\text{p}}}-\frac{1}{f_{\text{c}}}\right)-\frac{r}{z_{\text{p}}}\rho\cos(\theta-\phi)+\tilde{z}\left(\frac{f}{f^{\prime}}\rho,\theta\right)\right)\right].$
(C.26)

The terms proportional to $\rho$ and $\rho^{2}$ in the exponent are known as
distortion and curvature of field, respectively [111]. Both shift the 3D
position of the Gaussian focus888We remind the reader that this corresponds to
the position of maximum intensity for a point-like source, i.e. the center of
the Airy pattern. Distortion displaces this point horizontally, within the
focal plane where the Airy pattern is, and curvature of field vertically
compared to said plane. and the new position of maximum intensity is referred
to as the diffraction focus. Crucially, if only such terms exist, they do not
change the intensity profile around the diffraction focus compared to the
profile around the Gaussian focus for an aberration-free system. Therefore, we
do not need to set tolerances on these.

It is well known that for a polynomial aberration of the form
$\tilde{z}\propto\rho^{n}$, one can add aberrations of a lower order to
maximize the intensity at the Gaussian focus [111]. Because different
aberrations shift the diffraction focus by different amounts, it is convenient
to expand the aberration function in polynomials that have maximum intensity
at the same point and, in particular, at the Gaussian focus. We can directly
compare the spread of the image with the aberration-free Airy pattern. This
expansion is standard and is done in terms of Zernike polynomials, so that

$\tilde{\phi}^{\text{inst}}\left(\frac{f}{f^{\prime}}\rho,\theta\right)=\sum_{l,n,m}\epsilon_{nm}\alpha_{lnm}\bar{\lambda}R^{m}_{n}\left(\frac{\rho}{w/2}\right)\cos
m\theta,$ (C.27)

where $n\geq m$, $n-m$ is even, $\epsilon_{nm}=1/\sqrt{2}$ if $m=0$, $n\neq 0$
and $1$ otherwise, and $\alpha_{nlm}$ are dimensionless constants. Because
EPIC uses a broadband setup, it is useful to extract a reference length scale
$\bar{\lambda}$, which we choose to be $500$nm. The polynomials $R_{n}^{m}$
can be found in standard references [111]. Terms with $2l+m+n=4$ correspond to
Seidel or primary aberrations.

If the aberration is small and the induced phase $k\tilde{z}\lesssim 1$,
tolerances are derived in standard references [111]. If this phase is large,
which can be the case for intensity interferometry, we can use the stationary
phase approximation to evaluate the integrals of eq. C.26, by identifying the
critical points of the exponent. In fact, the precise functional form is not
needed, but rather only the area in which most of the intensity lies in.
Critical points will exist only for a finite range of $\rho$ and one can prove
analytically that the wave profile decays much faster outside this range. Our
criterion is that the pixel must be at least as large as the area in which
critical points exist. We have verified this numerically, as can be seen in
figure 22. The intensity lost using this criterion is $\mathcal{O}(1\%)$, but
looser requirements, such as the collection of half the intensity, will give
parametrically the same tolerances.

As an illustrative example, we will now calculate the tolerance on the primary
(Seidel) spherical aberration for the primary mirror.999Technically, Seidel
aberrations correspond to simple monomials, in contrast to the Zernike
polynomials. Because the Maréchal criterion for a well-corrected system refers
to the diffraction focus (that the rms wavefront error should be less than
$\lambda/14$), it is simpler to first derive tolerances of the Zernike
polynomial of the same order $n$ as the corresponding Seidel aberration, and
then match the coefficients [111]. Essentially identical calculations can be
carried out for primary coma and astigmatism. We collect our results in table
8.

For cylindrically symmetric Zernike polynomials $m=0$, the integral is
independent of $\phi$ and depends only on the ratio
$|\boldsymbol{\theta}_{\text{F}}|\equiv r/f_{\text{c}}$, so that

$E_{k}(|\boldsymbol{\theta}_{\text{F}}|,\phi)=-\frac{ike^{ikf_{\text{c}}}}{f_{\text{c}}}\int_{0}^{w/2}\text{d}\rho\,\rho
J_{0}\left(k|\boldsymbol{\theta}_{\text{F}}|\rho\right)\exp\left[ik\tilde{\phi}^{\text{inst}}\left(\frac{f}{f^{\prime}}\rho\right)\right].$
(C.28)

Since the spreading due to aberrations is going to be larger than the
diffraction limited spread
$|\boldsymbol{\theta}_{\text{F}}|\sim\lambda/w$,101010The physical angle of a
diffraction limited image is $\lambda/w$, where $w=f^{\prime}D/f$. This is not
in conflict with the fact that the diffraction limit of the telescope is
$\lambda/D$ since any relative angle on the sky will be magnified by
$f/f^{\prime}$. we can use the asymptotic expansion of the Bessel function and
then approximate the integral using the stationary phase approximation.

The $(n,m)=(4,0)$ Zernike polynomial $R_{4}^{0}(x)=6x^{4}-6x^{2}+1$
corresponds to the primary spherical aberration
$\tilde{z}(\rho)=\alpha_{4}^{\prime}\bar{\lambda}(\rho/(w/2))^{4}$. Critical
points exist only for $|\boldsymbol{\theta}_{F}|\leq
2(6/\sqrt{2})\alpha_{4,0}\bar{\lambda}/(w/2)\equiv\theta_{4,0}$. This we must
match with the angle subtended by a pixel of diameter $\Delta x_{\text{p}}$ at
Gaussian focus a distance $f_{\text{c}}$ from the camera lens. Rescaling
$\alpha_{4}$ to correspond to the Seidel coefficient $\alpha^{\prime}_{4}$, we
find

$\alpha^{\prime}_{4}\lesssim 50\left(\frac{\Delta
x_{\text{p}}}{20\mu\text{m}}\right)\left(\frac{100\text{mm}}{f_{\text{c}}}\right)\left(\frac{D}{10\text{m}}\right)\left(\frac{500\text{nm}}{\bar{\lambda}}\right)\left(\frac{10}{f/f^{\prime}}\right).$
(C.29)

This should be contrasted to the Maréchal criterion (the rms wavefront error
of a sphere centered at the diffraction focus should be smaller than
$\lambda/14$ for a diffraction limited system) which for a primary spherical
aberration yields $\alpha_{4}^{\prime}\bar{\lambda}\lesssim 0.94\lambda$. For
this choice of parameters, the rms wavefront error is $3.75\bar{\lambda}$,
which corresponds to twice the tolerable surface rms error for the primary
mirror111111 The rms wavefront error $\Delta\tilde{z}$ of the aberration
function $\tilde{z}$ is defined as
$(\Delta\tilde{z})^{2}\equiv\frac{1}{\pi}\int_{0}^{1}\int_{0}^{2\pi}\left(\tilde{z}-\left<\tilde{z}\right>\right)^{2}\rho\,\text{d}\rho\text{d}\theta,\quad\text{where}\quad\left<\tilde{z}\right>\equiv\frac{1}{\pi}\int_{0}^{1}\int_{0}^{2\pi}\tilde{z}\rho\,\text{d}\rho\text{d}\theta.$
(C.30) . As such, even in the presence of small pixels, intensity
interferometry can tolerate aberrations an order of magnitude larger than
diffraction limited designs. Because the surface rms error is about a micron,
rather than significantly sub-wavelength, many of the expensive and time-
consuming mirror polishing processes could be skipped [115]. Larger pixels,
allocation of multiple pixels to a single aberrated image and smaller
magnifications can all further loosen this tolerance.

Figure 22: Intensity plotted as a function of the physical pixel coordinate, where have normalized the intensities to the peak of the diffraction limited image. All curves correspond to the same total intensity. We have chosen a $D=10\,\mathrm{m}$ primary, $f/f^{\prime}=10$, $f_{\text{c}}=100\,$mm at $\lambda=500\,$nm. The lightest blue line corresponds to the diffraction limited image of a point source (Airy pattern), whose characteristic (magnified) diffraction limit $(f/f^{\prime})(\bar{\lambda/D})f_{\text{c}}$ we have indicated with the leftmost vertical dotted line. The line labelled “Instrumental” corresponds to the produced image in a system with primary spherical aberration $\alpha_{4}^{\prime}=50$. We have indicated the boundary of the region within which most of the intensity resides as $\theta_{4,0}f_{\text{c}}$, which agrees with the analytic estimate described in the main text, see e.g. C.29. The line titled “Atmospheric” is the time-averaged image of a point source after accounting for atmospheric fluctuations, as described by eq. C.31. We have chosen a Fried parameter $r_{0}=10$cm, which corresponds to a $1$as seeing disk at $500$nm, which we have indicated with the middlemost dotted line. The Maréchal criterion for a diffraction limited system imposes that the maximum intensity of the distorted image (normalized to the diffraction limited peak) be $\geq 0.8$, which shows how much more distortion EPIC can tolerate. Type of aberration | Form of $\tilde{z}(\rho,\theta)$ | Tolerance |  Wavefront error (rms) |  Maréchal criterion
---|---|---|---|---
Spherical aberration | $\alpha_{4}^{\prime}\bar{\lambda}\left(\frac{\rho}{D/2}\right)^{4}$ | $\alpha_{4}^{\prime}\lesssim 50$ | $\sim 3.8\bar{\lambda}$ | $\alpha_{4}^{\prime}\bar{\lambda}\lesssim 0.94\lambda$
Coma | $\alpha_{3,1}^{\prime}\bar{\lambda}\left(\frac{\rho}{D/2}\right)^{3}\cos\theta$ | $\alpha_{3,1}^{\prime}\lesssim 43$ | $\sim 5\bar{\lambda}$ | $\alpha_{3,1}^{\prime}\bar{\lambda}\lesssim 0.60\lambda$
Astigmatism | $\alpha_{2,2}^{\prime}\bar{\lambda}\left(\frac{\rho}{D/2}\right)^{2}\cos^{2}\theta$ | $\alpha_{2,2}^{\prime}\lesssim 100$ | $\sim 20\bar{\lambda}$ | $\alpha_{2,2}^{\prime}\bar{\lambda}\lesssim 0.35\lambda$
Curvature of field | $\alpha_{2}\bar{\lambda}\left(\frac{\rho}{D/2}\right)^{2}$ | - | - | -
Distortion | $\alpha_{1,1}\bar{\lambda}\left(\frac{\rho}{D/2}\right)\cos\theta$ | - | - | -
Table 8: Tolerances on Seidel aberrations assuming that all the intensity of a
single spectral channel is recorded by a single pixel. In the third column the
tolerance is set by imposing that all the intensity (see text for a precise
definition) falls within a pixel of linear size of $20\mu\,$m, for a telescope
with $f/f^{\prime}=10$, a $D=10\,$m primary and a $f_{\text{c}}=100\,$mm
camera focal length, at the reference wavelength of $\bar{\lambda}=500\,$nm.
The fourth column relates the tolerance to the rms wavefront error or,
equivalently, to twice the rms surface error of the primary mirror. In the
last column we give the corresponding Maréchal criterion for a diffraction
limited system from [111] for comparison. Larger pixels, allocation of more
pixels per spectral channel and smaller magnifications can further alleviate
these requirements.

#### Atmospheric aberrations

The atmosphere is another source of image aberrations that broadens the area
into which the light is focused on the photodetector. We have established in
sections 2.5 and B.2 that sources within the isoplanatic angle suffer the same
stochastic image variations from the atmosphere, so that the contrast of the
main EPIC fringe is not diminished. Here we are concerned with the size of the
distorted image over long timescales, which is known as the seeing disk. As
with the instrumental aberrations discussed above, the seeing disk should fit
within each pixel.

Following the modeling of atmospheric fluctuations based on Kolmogorov
turbulence, the intensity on the focal plane over long timescales is given by
[207]

$I(\boldsymbol{\theta}_{\text{F}})\propto\int\text{d}^{2}x\,e^{-ik\boldsymbol{\theta}_{\text{F}}\cdot\mathbb{x}}e^{-3.44\left(\frac{f}{f^{\prime}}\frac{|\mathbb{x}|}{r_{0}}\right)^{5/3}},$
(C.31)

where $r_{0}$ is the Fried parameter. The FWHM of the seeing disk is found to
be $\theta_{\text{seeing}}\sim 0.98\lambda/r_{0}$, which is given by eq. 2.63.
Here the seeing angle is magnified at the plane of the photodetector by
$f/f^{\prime}$. The Fried parameter $r_{0}\sim 10-20$cm, which corresponds to
a (unmagnified) seeing disk of $\sim 0.5-1$ arcsec for blue light. We plot the
radial profile in figure 22, to compare it with the spread due to instrumental
aberrations. In particular, the size of the magnified seeing disk is the same
as that of an image with primary spherical aberration for
$\alpha_{4}^{\prime}\sim 25$.

###  C.4 Dispersive element

In this section we fill in the details of the calculation presented in section
3.4, concerning the reduction of fringe contrast for large telescopes, large
pixels and fast photodetectors. We stress that the presence of aberrations,
instrumental or atmospheric, do not modify this effect. We thus study a
diffraction limited system here for clarity.

The analysis here is one-dimensional and the optical configuration consists of
a grating of size $Nd_{\text{g}}$, a camera lens of focal length
$f_{\text{c}}$ at a distance $z_{\text{l}}$ from the grating, and a
photodetector array at the focal plane of the lens. A plane monochromatic wave
incident on the grating has a profile at the focal plane given by

$E_{k}(\theta_{\text{F}},t)\propto
A_{k}e^{ik(f_{\text{c}}+z_{\text{l}}-ct)}\exp\left[ik\frac{x_{\text{F}}^{2}}{f_{\text{c}}+z_{\text{l}}}\right]\frac{\sin\left(k\theta_{\text{F}}\frac{Nd}{2}\right)}{\sin\left(k\theta_{\text{F}}\frac{d}{2}\right)}\frac{\sin\left(k\frac{s}{2}\theta_{\text{F}}\right)}{k\frac{s}{2}\theta_{\text{F}}}e^{-ikct},$
(C.32)

where $\theta_{\text{F}}\equiv x_{\text{F}}/f_{\text{c}}$ is the ratio of the
physical distance on the pixel plane over the camera focal length,
$d_{\text{g}}$ is the periodicity of the grating and $s$ is the width of each
slit. The factor $f_{\text{c}}+z_{\text{l}}$ is common for every wavelength
and can be absorbed into a shift of the clock, so will be neglected below.
Similarly, the quadratic in $x_{\text{F}}$ term is a path difference for rays
with different wavelengths due to being focused at different points. We will
neglect this for now but will comment on it towards the end. These phase
factors were neglected in section 3.4 for brevity.

We are concerned with the spatial distribution of different wavelengths within
a pixel, which comes about from the term of the form $\sin(Ny)/\sin(y)$. We
specialize to the pixel in whose center the focused wavenumber is $k=\bar{k}$.
All other wavelengths within the pixel will have wavenumbers
$k\equiv\bar{k}+\sigma\in[\bar{k}-\frac{\sigma_{k}}{2},\bar{k}+\frac{\sigma_{k}}{2}]$,
where $\sigma_{k}\sim\bar{k}/\mathcal{R}\ll\bar{k}$, $\mathcal{R}$ being the
required resolution. The first diffraction peak of the wavelength
$\lambda=2\pi/k$ occurs at $\theta_{k}=\frac{\lambda}{d_{\text{g}}}$. We can
shift $\theta_{\text{F}}$ such that
$\theta_{\text{F}}=\theta_{\text{F}}^{\prime}-\frac{\bar{\lambda}}{d}$,
bringing the pixel’s center at $\theta_{\text{F}}^{\prime}=0$, where
$\bar{\lambda}$ is focused at. Then
$k\theta_{\text{F}}\frac{Nd_{\text{g}}}{2}=k\theta_{\text{F}}^{\prime}\frac{Nd_{\text{g}}}{2}+\pi
N\frac{k}{\bar{k}}$ and similarly for the denominator. Now within the pixel of
interest
$\theta_{\text{F}}^{\prime}\in\left(-\frac{\bar{\lambda}}{2d_{\text{g}}\mathcal{R}},\frac{\bar{\lambda}}{2d_{\text{g}}\mathcal{R}}\right)$,
so that $k\theta_{\text{F}}^{\prime}d_{\text{g}}\ll 1$ and we can approximate
the denominator as
$\sin\left(k\theta_{\text{F}}\frac{d_{\text{g}}}{2}\right)=-\sin(k\theta_{\text{F}}\frac{d_{\text{g}}}{2}-\pi)=-\sin\left[k\theta_{\text{F}}^{\prime}\frac{d_{\text{g}}}{2}+\pi\left(\frac{k}{\bar{k}}-1\right)\right]\simeq-\left[k\theta_{\text{F}}^{\prime}\frac{d_{\text{g}}}{2}+\pi\left(\frac{k}{\bar{k}}-1\right)\right]$.
Similarly the numerator is
$-\sin\left[k\theta_{\text{F}}^{\prime}\frac{Nd_{\text{g}}}{2}+\pi
N\left(\frac{k}{\bar{k}}-1\right)\right]$. As proven in the beginning of
section C.3, the resulting profile is the image of the point source at the
wavelength $k$, as if we had an aperture of size $Nd_{\text{g}}$ without the
grating. Dropping the prime on $\theta_{\text{F}}^{\prime}$ from here on, we
recover eq. 3.7:

$E_{k}(\theta_{\text{F}},t)\propto
A_{k}\frac{\sin\left[k\theta_{\text{F}}\frac{Nd_{\text{g}}}{2}+\pi
N\left(\frac{k}{\bar{k}}-1\right)\right]}{k\theta_{\text{F}}\frac{Nd_{\text{g}}}{2}+\pi
N\left(\frac{k}{\bar{k}}-1\right)}e^{-ikct}.$ (3.7)

A straightforward calculation gives the numerator and denominator of the
excess fractional intensity correlation

$\begin{split}\mathcal{N}\propto&\left<\int\text{d}k\int\text{d}k^{\prime}|A_{k}|^{2}|A_{k^{\prime}}|^{2}e^{-i(k-k^{\prime})ct}\right.\\\
&\times\int\text{d}\theta_{\text{F}}^{(1)}\,N\frac{\sin\left[k\theta_{\text{F}}^{(1)}\frac{Nd_{\text{g}}}{2}+\pi
N\left(\frac{k}{\bar{k}}-1\right)\right]}{k\theta_{\text{F}}^{(1)}\frac{Nd_{\text{g}}}{2}+\pi
N\left(\frac{k}{\bar{k}}-1\right)}N\frac{\sin\left[k^{\prime}\theta_{\text{F}}^{(1)}\frac{Nd_{\text{g}}}{2}+\pi
N\left(\frac{k^{\prime}}{\bar{k}}-1\right)\right]}{k^{\prime}\theta_{\text{F}}^{(1)}\frac{Nd_{\text{g}}}{2}+\pi
N\left(\frac{k^{\prime}}{\bar{k}}-1\right)}\\\
&\left.\times\int\text{d}\theta_{\text{F}}^{(2)}\,N\frac{\sin\left[k\theta_{\text{F}}^{(2)}\frac{Nd_{\text{g}}}{2}+\pi
N\left(\frac{k}{\bar{k}}-1\right)\right]}{k\theta_{\text{F}}^{(2)}\frac{Nd_{\text{g}}}{2}+\pi
N\left(\frac{k}{\bar{k}}-1\right)}N\frac{\sin\left[k^{\prime}\theta_{\text{F}}^{(2)}\frac{Nd_{\text{g}}}{2}+\pi
N\left(\frac{k^{\prime}}{\bar{k}}-1\right)\right]}{k^{\prime}\theta_{\text{F}}^{(2)}\frac{Nd_{\text{g}}}{2}+\pi
N\left(\frac{k^{\prime}}{\bar{k}}-1\right)}\right>_{\sigma_{t}}\\\
\mathcal{D}\propto&\left|\int\text{d}\theta_{\text{F}}\int\text{d}k\,\left|A_{k}\frac{\sin\left[k\theta_{\text{F}}\frac{Nd_{\text{g}}}{2}+\pi
N\left(\frac{k}{\bar{k}}-1\right)\right]}{k\theta_{\text{F}}\frac{Nd_{\text{g}}}{2}+\pi
N\left(\frac{k}{\bar{k}}-1\right)}\right|^{2}\right|^{2},\end{split}$ (C.33)

where
$\mathcal{N}\equiv\left<I_{1}I_{2}\right>-\left<I_{1}\right>\left<I_{2}\right>$
and $\mathcal{D}\equiv\left<I_{1}\right>\left<I_{2}\right>$, as in eqs. 3.8
and 3.9, and $\left<...\right>_{\sigma_{t}}$ denotes averaging over time with
a normalized gaussian kernel of standard deviation $\sigma_{t}$. We emphasize
that the terms involving the focal plane of each telescope,
$\theta_{\text{F}}^{(i)}$, $i=1,2$ contain _different_ wavenumber $k$ and
$k^{\prime}$ in the numerator. Consequently, if these are focused too far
apart spatially, then this term averages out and fringe contrast can be lost.

There is no obstacle in evaluating these formulae numerically, but we can
understand the source of the suppression better in the limit where the pixel
is much larger than the width of these sinc functions (Airy patterns). Then we
can take the integrations over $\theta_{\text{F}}^{(i)}$ to infinity in these
expressions, which is a good approximation if there are several distinct “Airy
patterns” within each pixel. We have also verified this fact numerically.

One can show that

$\begin{split}\int_{-\infty}^{+\infty}\text{d}\theta_{\text{F}}^{(i)}\,&N\frac{\sin\left[k\theta_{\text{F}}^{(i)}\frac{Nd_{\text{g}}}{2}+\pi
N\left(\frac{k}{\bar{k}}-1\right)\right]}{k\theta_{\text{F}}^{(i)}\frac{Nd_{\text{g}}}{2}+\pi
N\left(\frac{k}{\bar{k}}-1\right)}N\frac{\sin\left[k^{\prime}\theta_{\text{F}}^{(i)}\frac{Nd_{\text{g}}}{2}+\pi
N\left(\frac{k^{\prime}}{\bar{k}}-1\right)\right]}{k^{\prime}\theta_{\text{F}}^{(i)}\frac{Nd_{\text{g}}}{2}+\pi
N\left(\frac{k^{\prime}}{\bar{k}}-1\right)}\\\ &\simeq\frac{2\pi
N}{\bar{k}d_{\text{g}}}\frac{\sin\left(N\pi\frac{k-k^{\prime}}{\bar{k}}\right)}{N\pi\frac{k-k^{\prime}}{\bar{k}}}.\end{split}$
(C.34)

This should be contrasted to the term $\left<I_{1}\right>\left<I_{2}\right>$,
which involves integrals of the same wavelengths at each telescope, in which
case $\left<I_{1}\right>=\left<I_{2}\right>\propto 2\pi
N/{\bar{k}d_{\text{g}}}$. Neglecting this prefactor, we have

$\begin{split}\mathcal{N}\propto\left<\int\text{d}k\int\text{d}k^{\prime}|A_{k}|^{2}|A_{k^{\prime}}|^{2}e^{-i(k-k^{\prime})ct}\left[\frac{\sin\left(N\pi\frac{k-k^{\prime}}{\bar{k}}\right)}{N\pi\frac{k-k^{\prime}}{\bar{k}}}\right]^{2}\right>_{\sigma_{t}}.\end{split}$
(C.35)

This shows where the “sinc-squared filter” we encountered in eq. 3.10 stems
from. It restricts the integral only to those wavenumbers that are
$\sim(N\pi)^{-1}$ close. Because $N\gg\mathcal{R}$ in order to accommodate
optical aberrations but, most importantly, seeing fluctuations, this term, if
uncorrected, will reduce the fringe contrast.

The starlight spectrum has a natural width $\sigma_{n}$ and a peak at $k_{0}$.
By writing
$|A_{k}|^{2}=\frac{1}{\sqrt{2\pi}\sigma_{n}}\exp\left[-\frac{(k-k_{0})^{2}}{2\sigma_{n}^{2}}\right]$,
where $\sigma_{n}\sim k_{0}\gg\sigma_{k}$, we can approximate within the
integral $|A_{k}|^{2}\simeq|A_{\bar{k}}|^{2}$, which is a constant and can be
dropped in ratios. We find:

$\begin{split}\frac{\mathcal{N}}{\mathcal{D}}\simeq\left<\int_{\bar{k}-\frac{\sigma_{k}}{2}}^{\bar{k}+\frac{\sigma_{k}}{2}}\text{d}k\int_{\bar{k}-\frac{\sigma_{k}}{2}}^{\bar{k}+\frac{\sigma_{k}}{2}}\text{d}k^{\prime}\,e^{-i(k-k^{\prime})ct}\left[\frac{\sin\left(N\pi\frac{k-k^{\prime}}{\bar{k}}\right)}{N\pi\frac{k-k^{\prime}}{\bar{k}}}\right]^{2}\right>_{\sigma_{t}}.\end{split}$
(C.36)

In the limit of no dispersion (such as in the case where we used a filter of
bandwidth $\sim\bar{k}/\mathcal{R}$ instead), this becomes

$\left<\int_{\bar{k}-\frac{\sigma_{k}}{2}}^{\bar{k}+\frac{\sigma_{k}}{2}}\text{d}k\int_{\bar{k}-\frac{\sigma_{k}}{2}}^{\bar{k}+\frac{\sigma_{k}}{2}}\text{d}k^{\prime}\,e^{-i(k-k^{\prime})ct}\right>_{\sigma_{t}}=\frac{1}{c^{2}\sigma_{t}^{2}}\left[-2+2e^{-c^{2}\sigma_{k}^{2}\sigma_{t}^{2}/2}+\sqrt{2\pi}c\sigma_{k}\sigma_{t}\,\text{erf}\left(\frac{c\sigma_{k}\sigma_{t}}{\sqrt{2}}\right)\right],$
(C.37)

which, in the limit $c\sigma_{t}\ll\sigma_{k}^{-1}$, along with the prefactor
$\propto\sigma_{n}^{-2}$ of the $|A_{k}|^{2}$ terms, gives
$\propto\left(\frac{\sigma_{k}}{\sigma_{n}}\right)^{2}\frac{1}{c\sigma_{k}\sigma_{t}}$.
The first quadratic term is a suppression coming from taking only part of the
total spectrum within the pixel, and the second term is the usual suppression
due to a finite time-resolution. This is the standard result used throughout
section 2.

The integral in the presence of dispersion cannot be calculated analytically,
but we can bring it in a form that is more transparent. We define the function
$h(\sigma)=e^{-i\sigma
ct}\left[\frac{\sin\left(\sigma/\sigma_{d}\right)}{\sigma/\sigma_{d}}\right]^{2}$,
where we defined $\sigma_{d}\equiv\bar{k}/{\pi N}$ as the “diffraction
limited” width. By changing variables to $\sigma\equiv k+k^{\prime}$ and
$\sigma^{\prime}\equiv k-k^{\prime}$, the double integral can be made one-
dimensional

$\begin{split}\left<\int_{\bar{k}-\frac{\sigma_{k}}{2}}^{\bar{k}+\frac{\sigma_{k}}{2}}\text{d}k\int_{\bar{k}-\frac{\sigma_{k}}{2}}^{\bar{k}+\frac{\sigma_{k}}{2}}\text{d}k^{\prime}\,e^{-i(k-k^{\prime})ct}\left[\frac{\sin\left(N\pi\frac{k-k^{\prime}}{\bar{k}}\right)}{N\pi\frac{k-k^{\prime}}{\bar{k}}}\right]^{2}\right>_{\sigma_{t}}&=\left<\int_{0}^{\sigma_{k}}\text{d}\sigma\,(\sigma_{k}-\sigma)\left[h(\sigma)+h(-\sigma)\right]\right>_{\sigma_{t}}\\\
=2&\int_{0}^{\sigma_{k}}\text{d}\sigma\,(\sigma_{k}-\sigma)e^{-\frac{c^{2}\sigma^{2}\sigma_{t}^{2}}{2}}\left[\frac{\sin\left(\sigma/\sigma_{d}\right)}{\sigma/\sigma_{d}}\right]^{2},\end{split}$
(C.38)

which concludes the proof of eq. 3.10. This function is plotted in figure 12
for relevant parameters of the different proposed phases of EPIC
interferometers. We refer the reader to section 3.4 for the implications of
these results and for potential ways to alleviate the resulting suppression.

We have carried out the same calculation in the presence of a prism instead of
a grating. The result remains the same as that of eq. C.33, with the only
difference that the quantity $N$ in the $\theta_{\text{F}}$-independent term
(the resolving power if the system was diffraction limited) is replaced by
$N_{\text{prism}}\equiv b\,\text{d}n/\text{d}\lambda$, where $b$ is the length
of the base of the prism and $\text{d}n/\text{d}\lambda$ the derivative of the
index of refraction with respect to the wavelength. The requirement of
including the seeing angle within each pixel is the same as in eq. 3.11, but
with $N\to N_{\text{prism}}$.

## References

  * [1] Ken Van Tilburg, Masha Baryakhtar, Marios Galanis, and Neal Weiner.  Astrometry with Extended-Path Intensity Correlation.  arXiv e-prints, page arXiv:2307.03221, July 2023.
  * [2] Kailash C. Sahu et al.  An Isolated Stellar-mass Black Hole Detected through Astrometric Microlensing.  Astrophys. J., 933(1):83, 2022.
  * [3] Casey Y. Lam et al.  An Isolated Mass-gap Black Hole or Neutron Star Detected with Astrometric Microlensing.  ApJ, 933(1):L23, July 2022.
  * [4] Przemek Mróz, Andrzej Udalski, and Andrew Gould.  Systematic Errors as a Source of Mass Discrepancy in Black Hole Microlensing Event OGLE-2011-BLG-0462.  ApJ, 937(2):L24, October 2022.
  * [5] Gaia Collaboration.  Gaia Data Release 3: Stellar multiplicity, a teaser for the hidden treasure.  arXiv e-prints, page arXiv:2206.05595, June 2022.
  * [6] Ana Bonaca, David W. Hogg, Adrian M. Price-Whelan, and Charlie Conroy.  The Spur and the Gap in GD-1: Dynamical evidence for a dark substructure in the Milky Way halo.  The Astrophysical Journal, 11 2019.
  * [7] Lina Necib, Bryan Ostdiek, Mariangela Lisanti, Timothy Cohen, Marat Freytsis, and Shea Garrison-Kimmel.  Chasing Accreted Structures within Gaia DR2 using Deep Learning.  Astrophys. J., 903(1):25, 2020.
  * [8] Amina Helmi.  Streams, Substructures, and the Early History of the Milky Way.  ARA&A, 58:205–256, August 2020.
  * [9] Amina Helmi.  Streams, Substructures, and the Early History of the Milky Way.  Ann. Rev. Astron. Astrophys., 58(1):205–256, 2020.
  * [10] John David Anderson, Michael SW Keesey, Eunice L Lau, EM Standish Jr, and XX Newhall.  Tests of general relativity using astrometric and radio metric observations of the planets.  Acta Astronautica, 5(1-2):43–61, 1978.
  * [11] E Fomalont, S Kopeikin, G Lanyi, and J Benson.  Progress in measurements of the gravitational bending of radio waves using the vlba.  The Astrophysical Journal, 699(2):1395, 2009.
  * [12] Maria Teresa Crosta and F Mignard.  Microarcsecond light bending by jupiter.  Classical and Quantum Gravity, 23(15):4853, 2006.
  * [13] GRAVITY Collaboration, R. Abuter, A. Amorim, M. Bauböck, J. P. Berger, H. Bonnet, W. Brandner, Y. Clénet, V. Coudé Du Foresto, P. T. de Zeeuw, C. Deen, J. Dexter, G. Duvert, A. Eckart, F. Eisenhauer, N. M. Förster Schreiber, P. Garcia, F. Gao, E. Gendron, R. Genzel, S. Gillessen, P. Guajardo, M. Habibi, X. Haubois, Th. Henning, S. Hippler, M. Horrobin, A. Huber, A. Jiménez-Rosales, L. Jocou, P. Kervella, S. Lacour, V. Lapeyrère, B. Lazareff, J. B. Le Bouquin, P. Léna, M. Lippa, T. Ott, J. Panduro, T. Paumard, K. Perraut, G. Perrin, O. Pfuhl, P. M. Plewa, S. Rabien, G. Rodríguez-Coira, G. Rousset, A. Sternberg, O. Straub, C. Straubmeier, E. Sturm, L. J. Tacconi, F. Vincent, S. von Fellenberg, I. Waisberg, F. Widmann, E. Wieprecht, E. Wiezorrek, J. Woillez, and S. Yazici.  Detection of orbital motions near the last stable circular orbit of the massive black hole SgrA*.  A&A, 618:L10, October 2018.
  * [14] Ken Van Tilburg, Anna-Maria Taki, and Neal Weiner.  Halometry from astrometry.  Journal of Cosmology and Astroparticle Physics, 2018(07):041, 2018\.
  * [15] Malte Buschmann, Benjamin R Safdi, and Katelin Schutz.  Galactic potential and dark matter density from angular stellar accelerations.  Physical review letters, 127(24):241104, 2021.
  * [16] Cristina Mondino, Anna-Maria Taki, Ken Van Tilburg, and Neal Weiner.  First Results on Dark Matter Substructure from Astrometric Weak Lensing.  Phys. Rev. Lett., 125(11):111101, 2020.
  * [17] Ted Pyne, Carl R Gwinn, Mark Birkinshaw, T Marshall Eubanks, and Demetrios N Matsakis.  Gravitational radiation and very long baseline interferometry.  arXiv preprint astro-ph/9507030, 1995.
  * [18] Laura G Book and Eanna E Flanagan.  Astrometric effects of a stochastic gravitational wave background.  Physical Review D, 83(2):024024, 2011.
  * [19] Sergei A Klioner.  Gaia-like astrometry and gravitational waves.  Classical and Quantum Gravity, 35(4):045005, 2018.
  * [20] Christopher J Moore, Deyan P Mihaylov, Anthony Lasenby, and Gerard Gilmore.  Astrometric search method for individually resolvable gravitational wave sources with gaia.  Physical Review Letters, 119(26):261102, 2017.
  * [21] E. Høg, C. Fabricius, V. V. Makarov, S. Urban, T. Corbin, G. Wycoff, U. Bastian, P. Schwekendiek, and A. Wicenec.  The Tycho-2 catalogue of the 2.5 million brightest stars.  A&A, 355:L27–L30, March 2000.
  * [22] Timo Prusti, JHJ De Bruijne, Anthony GA Brown, Antonella Vallenari, C Babusiaux, CAL Bailer-Jones, U Bastian, M Biermann, Dafydd Wyn Evans, L Eyer, et al.  The gaia mission.  Astronomy & astrophysics, 595:A1, 2016.
  * [23] A. Bradley, L. Abramowicz-Reed, D. Story, G. Benedict, and W. Jefferys.  The Flight Hardware and Ground System for Hubble Space Telescope Astrometry.  PASP, 103:317, March 1991.
  * [24] Jonathan P. Gardner, John C. Mather, Mark Clampin, Rene Doyon, Matthew A. Greenhouse, Heidi B. Hammel, John B. Hutchings, Peter Jakobsen, Simon J. Lilly, Knox S. Long, Jonathan I. Lunine, Mark J. McCaughrean, Matt Mountain, John Nella, George H. Rieke, Marcia J. Rieke, Hans-Walter Rix, Eric P. Smith, George Sonneborn, Massimo Stiavelli, H. S. Stockman, Rogier A. Windhorst, and Gillian S. Wright.  The James Webb Space Telescope.  Space Sci. Rev., 123(4):485–606, April 2006.
  * [25] Roberto Abuter, Matteo Accardo, A Amorim, N Anugu, Gerardo Avila, N Azouaoui, Myriam Benisty, Jean-Philippe Berger, Nicolas Blind, Henri Bonnet, et al.  First light for gravity: Phase referencing optical interferometry for the very large telescope interferometer.  Astronomy & Astrophysics, 602:A94, 2017.
  * [26] T. A. ten Brummelaar, H. A. McAlister, S. T. Ridgway, Jr. Bagnuolo, W. G., N. H. Turner, L. Sturmann, J. Sturmann, D. H. Berger, C. E. Ogden, R. Cadman, W. I. Hartkopf, C. H. Hopper, and M. A. Shure.  First Results from the CHARA Array. II. A Description of the Instrument.  ApJ, 628(1):453–465, July 2005.
  * [27] Kazunori Akiyama et al.  First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole.  Astrophys. J. Lett., 875:L1, 2019.
  * [28] Kazunori Akiyama et al.  First M87 Event Horizon Telescope Results. II. Array and Instrumentation.  Astrophys. J. Lett., 875(1):L2, 2019.
  * [29] M. Mark Colavita and Peter L. Wizinowich.  Keck Interferometer update.  In Wesley A. Traub, editor, Interferometry for Optical Astronomy II, volume 4838 of Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, pages 79–88, February 2003.
  * [30] R. Hanbury Brown and R. G. Twiss.  A New type of interferometer for use in radio astronomy.  Philosophical Magazine, 45(366):663–682, March 1954.
  * [31] R. Hanbury Brown and R. Q. Twiss.  Correlation between Photons in two Coherent Beams of Light.  Nature, 177(4497):27–29, January 1956.
  * [32] R. Hanbury Brown.  A Test of a New Type of Stellar Interferometer on Sirius.  Nature, 178(4541):1046–1048, November 1956.
  * [33] R Hanbury Brown and Richard Quinton Twiss.  Interferometry of the intensity fluctuations in light-i. basic theory: the correlation between photons in coherent beams of radiation.  Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 242(1230):300–324, 1957.
  * [34] R Hanbury Brown and RQ Twiss.  Interferometry of the intensity fluctuations in light. ii. an experimental test of the theory for partially coherent light.  Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 243(1234):291–319, 1958.
  * [35] R Hanbury Brown and RQ Twiss.  Interferometry of the intensity fluctuations in light iii. applications to astronomy.  Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 248(1253):199–221, 1958.
  * [36] R. Hanbury Brown and R. Q. Twiss.  Interferometry of the Intensity Fluctuations in Light. IV. A Test of an Intensity Interferometer on Sirius A.  Proceedings of the Royal Society of London Series A, 248(1253):222–237, November 1958.
  * [37] R. Hanbury Brown, J. Davis, and L. R. Allen.  The stellar interferometer at Narrabri Observatory I. A. description of the instrument and the observational procedure.  MNRAS, 137:375, January 1967.
  * [38] R. Hanbury Brown, J. Davis, L. R. Allen, and J. M. Rome.  The stellar interferometer at Narrabri Observatory-II. The angular diameters of 15 stars.  MNRAS, 137:393, January 1967.
  * [39] R. Hanbury Brown.  The intensity interferometer; its application to astronomy.  London, Taylor & Francis; New York, Halsted Press, 1974.
  * [40] R. Hanbury Brown, J. Davis, and L. R. Allen.  The Angular Diameters of 32 Stars.  MNRAS, 167:121–136, April 1974.
  * [41] A. D. Code, J. Davis, R. C. Bless, and R. H. Brown.  Empirical effective temperatures and bolometric corrections for early-type stars.  ApJ, 203:417–434, January 1976.
  * [42] D. Herbison-Evans, R. Hanbury Brown, J. Davis, and L. R. Allen.  A study of alpha Virginis with an intensity interferometer.  MNRAS, 151:161, 1971.
  * [43] TC Weekes, H Badran, SD Biller, I Bond, S Bradbury, J Buckley, D Carter-Lewis, M Catanese, S Criswell, W Cui, et al.  Veritas: the very energetic radiation imaging telescope array system.  Astroparticle Physics, 17(2):221–243, 2002.
  * [44] James Anthony Hinton, Hess Collaboration, et al.  The status of the hess project.  New Astronomy Reviews, 48(5-6):331–337, 2004.
  * [45] C Baixeras, D Bastieri, C Bigongiari, O Blanch, G Blanchot, R Bock, T Bretz, A Chilingarian, JA Coarasa, E Colombo, et al.  Commissioning and first tests of the magic telescope.  Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 518(1-2):188–192, 2004.
  * [46] H Anderhub, Michael Backes, A Biland, Vittorio Boccone, Isabel Braun, Thomas Bretz, J Buß, Franck Cadoux, V Commichau, L Djambazov, et al.  Design and operation of fact–the first g-apd cherenkov telescope.  Journal of Instrumentation, 8(06):P06008, 2013.
  * [47] CTA Consortium J. Knapp@ leeds. ac. uk http://www. cta-observatory. org, M Actis, G Agnetta, F Aharonian, A Akhperjanian, J Aleksić, E Aliu, D Allan, I Allekotte, F Antico, et al.  Design concepts for the cherenkov telescope array cta: an advanced facility for ground-based high-energy gamma-ray astronomy.  Experimental Astronomy, 32:193–316, 2011.
  * [48] Paul D Nuñez, Richard Holmes, David Kieda, Janvida Rou, and Stephan LeBohec.  Imaging submilliarcsecond stellar features with intensity interferometry using air cherenkov telescope arrays.  Monthly Notices of the Royal Astronomical Society, 424(2):1006–1011, 2012.
  * [49] D. Kieda, Monica Acosta, Anastasia Barbano, Colin Carlile, Michael Daniel, Dainis Dravins, Jamie Holder, Nolan Matthews, Teresa Montaruli, Roland Walter, and Luca Zampieri.  Science opportunities enabled by the era of Visible Band Stellar Imaging with sub-100 $\mu$arc-sec angular resolution.  arXiv e-prints, page arXiv:1908.03164, August 2019.
  * [50] Sascha Trippe, Jae-Young Kim, Bangwon Lee, Changsu Choi, Junghwan Oh, Taeseok Lee, Sung-Chul Yoon, Myungshin Im, and Yong-Sun Park.  Optical Multi-Channel Intensity Interferometry - Or: How to Resolve O-Stars in the Magellanic Clouds.  Journal of Korean Astronomical Society, 47(6):235–253, December 2014.
  * [51] Paul D Nunez and A Domiciano de Souza.  Capabilities of future intensity interferometers for observing fast-rotating stars: imaging with two-and three-telescope correlations.  Monthly Notices of the Royal Astronomical Society, 453(2):1999–2005, 2015.
  * [52] Dainis Dravins.  Intensity interferometry: optical imaging with kilometer baselines.  In Optical and Infrared Interferometry and Imaging V, volume 9907, pages 128–139. SPIE, 2016.
  * [53] N Matthews and S LeBohec.  Astrophysical measurements with the veritas stellar intensity interferometer.  arXiv preprint arXiv:1908.03587, 2019.
  * [54] AU Abeysekara, W Benbow, A Brill, JH Buckley, JL Christiansen, AJ Chromey, MK Daniel, J Davis, A Falcone, Q Feng, et al.  Demonstration of stellar intensity interferometry with the four veritas telescopes.  Nature Astronomy, 4(12):1164–1169, 2020.
  * [55] David Kieda, Jonathan Davis, Tugdual LeBohec, Mike Lisa, and Nolan K Matthews.  The veritas-stellar intensity interferometry (vsii) survey of stellar diameters.  arXiv preprint arXiv:2108.09774, 2021.
  * [56] Luca Zampieri, Giampiero Naletto, Aleksandr Burtovoi, Michele Fiori, and Cesare Barbieri.  Stellar intensity interferometry of vega in photon counting mode.  Monthly Notices of the Royal Astronomical Society, 506(2):1585–1594, 2021.
  * [57] Dainis Dravins, Stephan LeBohec, Hannes Jensen, and Paul D. Nuñez.  Stellar intensity interferometry: Prospects for sub-milliarcsecond optical imaging.  New A Rev., 56(5):143–167, November 2012.
  * [58] Elliott P Horch, Samuel A Weiss, Paul M Klaucke, Richard A Pellegrino, and Justin D Rupert.  Observations with the southern connecticut stellar interferometer. i. instrument description and first results.  The Astronomical Journal, 163(2):92, 2022.
  * [59] W. Guerin, J. P. Rivet, M. Fouché, G. Labeyrie, D. Vernet, F. Vakili, and R. Kaiser.  Spatial intensity interferometry on three bright stars.  MNRAS, 480(1):245–250, October 2018.
  * [60] Paul Stankus, Andrei Nomerotski, Anže Slosar, and Stephen Vintskevich.  Two-photon amplitude interferometry for precision astrometry.  10 2020.
  * [61] Zhi Chen, Andrei Nomerotski, Anže Slosar, Paul Stankus, and Stephen Vintskevich.  Astrometry in two-photon interferometry using an Earth rotation fringe scan.  Phys. Rev. D, 107(2):023015, 2023.
  * [62] Massimo Tinto and Sanjeev V Dhurandhar.  Time-delay interferometry.  Living Reviews in Relativity, 17:1–54, 2014.
  * [63] James D Franson.  Bell inequality for position and time.  Physical review letters, 62(19):2205, 1989.
  * [64] Gordon Baym.  The physics of hanbury brown–twiss intensity interferometry: from stars to nuclear collisions.  1998\.
  * [65] Leonard Mandel and Emil Wolf.  Optical coherence and quantum optics.  Cambridge university press, 1995.
  * [66] Antoine Labeyrie, Stephen G Lipson, and Peter Nisenson.  An introduction to optical stellar interferometry.  Cambridge University Press, 2006.
  * [67] Sebastian Karl, Andreas Zmija, Stefan Richter, Naomi Vogel, Dmitry Malyshev, Adrian Zink, Thilo Michel, Gisela Anton, Joachim von Zanthier, and Stefan Funk.  Comparing different approaches for stellar intensity interferometry.  Monthly Notices of the Royal Astronomical Society, 512(2):1722–1729, 2022.
  * [68] Manuel Bojer, Zixin Huang, Sebastian Karl, Stefan Richter, Pieter Kok, and Joachim von Zanthier.  A quantitative comparison of amplitude versus intensity interferometry for astronomy.  New Journal of Physics, 24(4):043026, 2022.
  * [69] GRAVITY Collaboration, R. Abuter, M. Accardo, A. Amorim, N. Anugu, G. Ávila, N. Azouaoui, M. Benisty, J. P. Berger, N. Blind, H. Bonnet, P. Bourget, W. Brandner, R. Brast, A. Buron, L. Burtscher, F. Cassaing, F. Chapron, É. Choquet, Y. Clénet, C. Collin, V. Coudé Du Foresto, W. de Wit, P. T. de Zeeuw, C. Deen, F. Delplancke-Ströbele, R. Dembet, F. Derie, J. Dexter, G. Duvert, M. Ebert, A. Eckart, F. Eisenhauer, M. Esselborn, P. Fédou, G. Finger, P. Garcia, C. E. Garcia Dabo, R. Garcia Lopez, E. Gendron, R. Genzel, S. Gillessen, F. Gonte, P. Gordo, M. Grould, U. Grözinger, S. Guieu, P. Haguenauer, O. Hans, X. Haubois, M. Haug, F. Haussmann, Th. Henning, S. Hippler, M. Horrobin, A. Huber, Z. Hubert, N. Hubin, C. A. Hummel, G. Jakob, A. Janssen, L. Jochum, L. Jocou, A. Kaufer, S. Kellner, S. Kendrew, L. Kern, P. Kervella, M. Kiekebusch, R. Klein, Y. Kok, J. Kolb, M. Kulas, S. Lacour, V. Lapeyrère, B. Lazareff, J. B. Le Bouquin, P. Lèna, R. Lenzen, S. Lévêque, M. Lippa, Y. Magnard, L. Mehrgan, M. Mellein, A. Mérand, J. Moreno-Ventas, T. Moulin, E. Müller, F. Müller, U. Neumann, S. Oberti, T. Ott, L. Pallanca, J. Panduro, L. Pasquini, T. Paumard, I. Percheron, K. Perraut, G. Perrin, A. Pflüger, O. Pfuhl, T. Phan Duc, P. M. Plewa, D. Popovic, S. Rabien, A. Ramírez, J. Ramos, C. Rau, M. Riquelme, R. R. Rohloff, G. Rousset, J. Sanchez-Bermudez, S. Scheithauer, M. Schöller, N. Schuhler, J. Spyromilio, C. Straubmeier, E. Sturm, M. Suarez, K. R. W. Tristram, N. Ventura, F. Vincent, I. Waisberg, I. Wank, J. Weber, E. Wieprecht, M. Wiest, E. Wiezorrek, M. Wittkowski, J. Woillez, B. Wolff, S. Yazici, D. Ziegler, and G. Zins.  First light for GRAVITY: Phase referencing optical interferometry for the Very Large Telescope Interferometer.  A&A, 602:A94, June 2017.
  * [70] N. I. Shakura and R. A. Sunyaev.  Black holes in binary systems. Observational appearance.  A&A, 24:337–355, January 1973.
  * [71] Nolan Matthews, Jean-Pierre Rivet, Mathilde Hugbart, Guillaume Labeyrie, Robin Kaiser, Olivier Lai, Farrokh Vakili, David Vernet, Julien Chabé, Clémont Courde, Nicolas Schuhler, Pierre Bourget, and William Guerin.  Intensity interferometry at Calern and beyond: progress report.  In Antoine Mérand, Stephanie Sallum, and Joel Sanchez-Bermudez, editors, Optical and Infrared Interferometry and Imaging VIII, volume 12183 of Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, page 121830G, August 2022.
  * [72] VB Mendes and Erricos C Pavlis.  High-accuracy zenith delay prediction at optical wavelengths.  Geophysical research letters, 31(14), 2004.
  * [73] GC Hulley and Erricos C Pavlis.  A ray-tracing technique for improving satellite laser ranging atmospheric delay corrections, including the effects of horizontal refractivity gradients.  Journal of Geophysical Research: Solid Earth, 112(B6), 2007.
  * [74] Benjamin K Stuhl.  Atmospheric refraction corrections in ground-to-satellite optical time transfer.  Optics Express, 29(9):13706–13714, 2021.
  * [75] A Kolmogorov.  Classic papers on statistical theory, friedlander s., topper l, 1961.
  * [76] The effects of the turbulent atmosphere on wave propagation.  Jerusalem: Israel Program for Scientific Translations, 1971, 1971\.
  * [77] F. Roddier.  The effects of atmospheric turbulence in optical astronomy.  Progess in Optics, 19:281–376, January 1981.
  * [78] Changhui Rao, Wenhan Jiang, and Ning Ling.  Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence.  Journal of Modern Optics, 47(6):1111–1126, June 2000.
  * [79] MG Miller and PL Zieske.  Turbulence environment characterization.  Technical report, AVCO EVERETT RESEARCH LAB INC EVERETT MA, 1979.
  * [80] David L Fried.  Anisoplanatism in adaptive optics.  JOSA, 72(1):52–61, 1982.
  * [81] Frederick G Smith.  Atmospheric propagation of radiation.  The infrared and electro-optical systems handbook-IR/EO systems handbook, 1993.
  * [82] W Gao, Seung-Woo Kim, H Bosse, H Haitjema, YL Chen, XD Lu, W Knapp, A Weckenmann, WT Estler, and H Kunzmann.  Measurement technologies for precision positioning.  CIRP annals, 64(2):773–796, 2015.
  * [83] Yuki Shimizu, Liang-Chia Chen, Dae Wook Kim, Xiuguo Chen, Xinghui Li, and Hiraku Matsukuma.  An insight into optical metrology in manufacturing.  Measurement Science and Technology, 32(4):042003, April 2021.
  * [84] Vanessa P. Bailey, Philip M. Hinz, Alfio T. Puglisi, Simone Esposito, Vidhya Vaitheeswaran, Andrew J. Skemer, Denis Defrère, Amali Vaz, and Jarron M. Leisenring.  Large binocular telescope interferometer adaptive optics: on-sky performance and lessons learned.  In Enrico Marchetti, Laird M. Close, and Jean-Pierre Vran, editors, Adaptive Optics Systems IV, volume 9148 of Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, page 914803, July 2014.
  * [85] P. Wizinowich, J. Chin, C. Correia, J. Lu, T. Brown, K. Casey, S. Cetre, J.-R. Delorme, L. Gers, L. Hunter, S. Lilley, S. Ragland, A. Surendran, E. Wetherell, A. Ghez, T. Do, T. Jones, M. Liu, D. Mawet, C. Max, M. Morris, T. Treu, and S. Wright.  Keck all sky precision adaptive optics.  In Laura Schreiber, Dirk Schmidt, and Elise Vernet, editors, Adaptive Optics Systems VII, volume 11448, page 114480E. International Society for Optics and Photonics, SPIE, 2020.
  * [86] MA Holdaway and Tamara T Helfer.  Interferometric array design.  In Synthesis Imaging in Radio Astronomy II, volume 180, page 537, 1999.
  * [87] Eric Keto.  The shapes of cross-correlation interferometers.  The Astrophysical Journal, 475(2):843, 1997.
  * [88] Marcel JE Golay.  Point arrays having compact, nonredundant autocorrelations.  JOSA, 61(2):272–273, 1971.
  * [89] L Kogan.  Optimization of an array configuration minimizing side lobes.  MMA Memo, 171, 1997.
  * [90] F Boone.  Interferometric array design: Distributions of fourier samples for imaging.  Astronomy & Astrophysics, 386(3):1160–1171, 2002.
  * [91] Lennart Lindegren, SA Klioner, J Hernández, A Bombrun, M Ramos-Lerate, Hea Steidelmüller, U Bastian, M Biermann, A de Torres, E Gerlach, et al.  Gaia early data release 3-the astrometric solution.  Astronomy & Astrophysics, 649:A2, 2021.
  * [92] Stephan Steinhauer, Samuel Gyger, and Val Zwiller.  Progress on large-scale superconducting nanowire single-photon detectors.  Applied Physics Letters, 118(10):100501, 2021.
  * [93] Robert H. Hadfield.  Superfast photon counting.  Nature Photonics, 14(4):201–202, 2020.
  * [94] Boris Korzh, Qing-Yuan Zhao, Jason P Allmaras, Simone Frasca, Travis M Autry, Eric A Bersin, Andrew D Beyer, Ryan M Briggs, Bruce Bumble, Marco Colangelo, et al.  Demonstration of sub-3 ps temporal resolution with a superconducting nanowire single-photon detector.  Nature Photonics, 14(4):250–255, 2020.
  * [95] E. E. Wollman, V. B. Verma, A. D. Beyer, R. M. Briggs, B. Korzh, J. P. Allmaras, F. Marsili, A. E. Lita, R. P. Mirin, S. W. Nam, and M. D. Shaw.  Uv superconducting nanowire single-photon detectors with high efficiency, low noise, and 4 k operating temperature.  Opt. Express, 25(22):26792–26801, Oct 2017.
  * [96] Lixing You, Xiaoyan Yang, Yuhao He, Wenxing Zhang, Dengkuan Liu, Weijun Zhang, Lu Zhang, Ling Zhang, Xiaoyu Liu, Sijing Chen, Zhen Wang, and Xiaoming Xie.  Jitter analysis of a superconducting nanowire single photon detector.  AIP Advances, 3(7):072135, 2013.
  * [97] Andrew S. Mueller, Boris Korzh, Marcus Runyan, Emma E. Wollman, Andrew D. Beyer, Jason P. Allmaras, Angel E. Velasco, Ioana Craiciu, Bruce Bumble, Ryan M. Briggs, Lautaro Narvaez, Cristián Pe na, Maria Spiropulu, and Matthew D. Shaw.  Free-space coupled superconducting nanowire single-photon detector with low dark counts.  Optica, 8(12):1586–1587, Dec 2021.
  * [98] Dileep V. Reddy, Robert R. Nerem, Sae Woo Nam, Richard P. Mirin, and Varun B. Verma.  Superconducting nanowire single-photon detectors with 98$\%$ system detection efficiency at 1550 nm.  Optica, 7(12):1649–1653, Dec 2020.
  * [99] Emma E. Wollman, Varun B. Verma, Adriana E. Lita, William H. Farr, Matthew D. Shaw, Richard P. Mirin, and Sae Woo Nam.  Kilopixel array of superconducting nanowire single-photon detectors.  Opt. Express, 27(24):35279–35289, Nov 2019.
  * [100] Jason P. Allmaras, Emma E. Wollman, Andrew D. Beyer, Ryan M. Briggs, Boris A. Korzh, Bruce Bumble, and Matthew D. Shaw.  Demonstration of a thermally coupled row-column snspd imaging array.  Nano Letters, 20(3):2163–2168, 2020.  PMID: 32091221.
  * [101] Qing-Yuan Zhao, Di Zhu, Niccolò Calandri, Andrew E. Dane, Adam N. McCaughan, Francesco Bellei, Hao-Zhu Wang, Daniel F. Santavicca, and Karl K. Berggren.  Single-photon imager based on a superconducting nanowire delay line.  Nature Photonics, 11(4):247–251, 2017.
  * [102] Emma E. Wollman, Varun B. Verma, Alexander B. Walter, Jeff Chiles, Boris Korzh, Jason P. Allmaras, Yao Zhai, Adriana E. Lita, Adam N. McCaughan, Ekkehart Schmidt, Simone Frasca, Richard P. Mirin, Sae-Woo Nam, and Matthew D. Shaw.  Recent advances in superconducting nanowire single-photon detector technology for exoplanet transit spectroscopy in the mid-infrared.  Journal of Astronomical Telescopes, Instruments, and Systems, 7(1):011004, 2021.
  * [103] Cara Battersby, Lee Armus, Edwin Bergin, Tiffany Kataria, Margaret Meixner, Alexandra Pope, Kevin B. Stevenson, Asantha Cooray, David Leisawitz, Douglas Scott, James Bauer, C. Matt Bradford, Kimberly Ennico, Jonathan J. Fortney, Lisa Kaltenegger, Gary J. Melnick, Stefanie N. Milam, Desika Narayanan, Deborah Padgett, Klaus Pontoppidan, Thomas Roellig, Karin Sandstrom, Kate Y. L. Su, Joaquin Vieira, Edward Wright, Jonas Zmuidzinas, Johannes Staguhn, Kartik Sheth, Dominic Benford, Eric E. Mamajek, Susan G. Neff, Sean Carey, Denis Burgarella, Elvire De Beck, Maryvonne Gerin, Frank P. Helmich, S. Harvey Moseley, Itsuki Sakon, and Martina C. Wiedner.  The origins space telescope.  Nature Astronomy, 2(8):596–599, 2018.
  * [104] Simone Tisa, Fabrizio Guerrieri, and Franco Zappa.  Variable-load quenching circuit for single-photon avalanche diodes.  Opt. Express, 16(3):2232–2244, Feb 2008.
  * [105] Mirko Sanzaro, Paolo Gattari, Federica Villa, Alberto Tosi, Giuseppe Croce, and Franco Zappa.  Single-photon avalanche diodes in a 0.16 $\mu$m bcd technology with sharp timing response and red-enhanced sensitivity.  IEEE Journal of Selected Topics in Quantum Electronics, 24(2):1–9, 2018.
  * [106] Wolfgang Becker, Francesco Gramuglia, Ming-Lo Wu, Edoardo Charbon, and Claudio Bruschini.  8.7 ps fwhm irf width from ultrafast spad.  https://www.becker-hickl.com/literature/application-notes/8-7-ps-fwhm-irf-width-from-ultrafast-spad/, 2022\.  [Online; accessed 06-Apr-2023].
  * [107] Brian F. Aull, Erik K. Duerr, Jonathan P. Frechette, K. Alexander McIntosh, Daniel R. Schuette, and Richard D. Younger.  Large-format geiger-mode avalanche photodiode arrays and readout circuits.  IEEE Journal of Selected Topics in Quantum Electronics, 24(2):1–10, 2018.
  * [108] Meera Punjiya, Kevin Ryu, Brian Aull, K. Alexander McIntosh, Kevan Donlon, Mike Brattain, Hermanus Pretorius, Denis Nothern, Noah Pestana, Thomas Karolyshyn, Jorgo Mihallari, Zachary Kranefeld, Erik Duerr, Anup B. Katake, and Alejandro Miguel San Martin.  A direct-detection LIDAR detector for the Europa lander concept.  In Mark A. Itzler, Joshua C. Bienfang, and K. Alex McIntosh, editors, Advanced Photon Counting Techniques XVI, volume 12089, page 1208903. International Society for Optics and Photonics, SPIE, 2022.
  * [109] Meera Punjiya, Kevin Ryu, Matt Gregory, Alex McIntosh, Erik Duerr, and Anup Katake.  Radiation-Hardened Sensors for the Final Descent Phase of Europa Lander.  In Advanced Technology for National Security Workshop (ATNS) 2021, Pasadena, California, April 6-8, 2021. Pasadena, CA: Jet Propulsion Laboratory, National Aeronautics and Space Administration, 2021.
  * [110] Kevin K. Ryu, Brian F. Aull, Michael Collins, Noah Pestana, Joseph Ciampi, Rabindra Das, Kevan Donlon, Hermanus Pretorius, Meera Punjiya, Alex McIntosh, and Erik K Duerr.  Geiger-mode avalanche photodiode arrays fabricated on SOI engineered-substrates.  In Mark A. Itzler, Joshua C. Bienfang, and K. Alex McIntosh, editors, Advanced Photon Counting Techniques XVI, volume 12089, page 120890C. International Society for Optics and Photonics, SPIE, 2022.
  * [111] Max Born, Emil Wolf, A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock.  Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light.  Cambridge University Press, 7 edition, 1999.
  * [112] Pierre-Marie Gori, Farrokh Vakili, Jean-Pierre Rivet, William Guerin, Mathilde Hugbart, Andrea Chiavassa, Adrien Vakili, Robin Kaiser, and Guillaume Labeyrie.  I3T: Intensity Interferometry Imaging Telescope.  MNRAS, 505(2):2328–2335, August 2021.
  * [113] N. Jovanovic, C. Schwab, N. Cvetojevic, O. Guyon, and F. Martinache.  Enhancing Stellar Spectroscopy with Extreme Adaptive Optics and Photonics.  PASP, 128(970):121001, December 2016.
  * [114] Rüdiger Paschotta.  Encyclopedia of laser physics and technology.  John Wiley & Sons, 2008.
  * [115] Pierre Bely.  The design and construction of large optical telescopes.  Springer, 2003.
  * [116] Ed B. Fomalont.  Astrometry and Geodesy.  In G. B. Taylor, C. L. Carilli, and R. A. Perley, editors, Synthesis Imaging in Radio Astronomy II, volume 180 of Astronomical Society of the Pacific Conference Series, page 463, January 1999.
  * [117] A Richard Thompson, James M Moran, and George W Swenson.  Interferometry and synthesis in radio astronomy.  Springer Nature, 2017.
  * [118] Hayo Hase, Dirk Behrend, Chopo Ma, Bill Petrachenko, Harald Schuh, and Alan Whitney.  The emerging vgos network of the ivs.  In IVS 2012 General Meeting Proceedings, NASA/CP-2012-217504, pages 8–12, 2012.
  * [119] Dirk Behrend, Bill Petrachenko, Chet Ruszczyk, and Pedro Elósegui.  Roll-out status of the vgos network.  In poster, 24rd European VLBI group for geodesy and astrometry working meeting, 2019.
  * [120] A Niell, J Barrett, A Burns, R Cappallo, B Corey, M Derome, C Eckert, P Elosegui, R McWhirter, M Poirier, et al.  Demonstration of a broadband very long baseline interferometer system: a new instrument for high-precision space geodesy.  Radio Science, 53(10):1269–1291, 2018.
  * [121] María J Rioja and Richard Dodson.  Precise radio astrometry and new developments for the next generation of instruments.  The Astronomy and Astrophysics Review, 28:1–79, 2020.
  * [122] Microchip Technology Inc.  Mhm-2020 active hydrogen maser.  https://ww1.microchip.com/downloads/aemDocuments/documents/FTD/ProductDocuments/Brochures/MHM-2020-Active-Hydrogen-Maser-00003090.pdf, 2023\.  [Online; accessed 29-Jan-2023].
  * [123] Quartzlock.  Ch1-76a passive hydrogen maser.  https://www.quartzlock.com/userfiles/downloads/datasheets/Passive_Hydrogen_Maser_CH1-76A.pdf, 2023\.  [Online; accessed 29-Jan-2023].
  * [124] Microchip Technology Inc.  8040c rubidium frequency standard.  https://ww1.microchip.com/downloads/aemDocuments/documents/FTD/ProductDocuments/Brochures/00003047.pdf, 2023\.  [Online; accessed 29-Jan-2023].
  * [125] Astronic Corporation.  3352a series rubidium frequency standard.  https://www.astronics.com/docs/default-source/Documents-ats/current-datasheets/rubidiums/3352a_0.pdf?sfvrsn=2, 2023.  [Online; accessed 29-Jan-2023].
  * [126] Microchip Technology Inc.  5071a primary frequency standard.  https://ww1.microchip.com/downloads/aemDocuments/documents/FTD/ProductDocuments/Brochures/5071A-Sell-Sheet-00002980.pdf, 2023\.  [Online; accessed 29-Jan-2023].
  * [127] Anthony G. A. Brown.  Microarcsecond Astrometry: Science Highlights from Gaia.  ARA&A, 59:59–115, September 2021.
  * [128] Ken Van Tilburg and David E Kaplan.  Discovering subsolar dark matter halos with astrometric weak lensing of muliply-imaged quasars.  In preparation, 2023.
  * [129] Rex A Saffer, James Liebert, and Edward W Olszewski.  Discovery of a close detached binary da white dwarf system.  The Astrophysical Journal, 334:947–957, 1988.
  * [130] P Bergeron, F Wesemael, James Liebert, and G Fontaine.  Determination of the atmospheric parameters of the binary da white dwarf l870-2 (eg 11).  The Astrophysical Journal, 345:L91–L94, 1989.
  * [131] R Napiwotzki, CA Karl, T Lisker, S Catalán, H Drechsel, U Heber, D Homeier, D Koester, B Leibundgut, TR Marsh, et al.  The eso supernovae type ia progenitor survey (spy)-the radial velocities of 643 da white dwarfs.  Astronomy & Astrophysics, 638:A131, 2020.
  * [132] PFL Maxted, TR Marsh, and CKJ Moran.  Radial velocity measurements of white dwarfs.  Monthly Notices of the Royal Astronomical Society, 319(1):305–317, 2000.
  * [133] Jerry B Holberg, Terry D Oswalt, Edward M Sion, and George P McCook.  The 25 parsec local white dwarf population.  Monthly Notices of the Royal Astronomical Society, 462(3):2295–2318, 2016.
  * [134] P Bergeron, P Dufour, G Fontaine, S Coutu, S Blouin, C Genest-Beaulieu, A Bédard, and B Rolland.  On the measurement of fundamental parameters of white dwarfs in the gaia era.  The Astrophysical Journal, 876(1):67, 2019.
  * [135] Vedant Chandra, Hsiang-Chih Hwang, Nadia L Zakamska, and Sihao Cheng.  A gravitational redshift measurement of the white dwarf mass–radius relation.  The Astrophysical Journal, 899(2):146, 2020.
  * [136] JA Panei, LG Althaus, and OG Benvenuto.  Mass-radius relations for white dwarf stars of different internal compositions.  Astron. Astrophys, 353:970–977, 2000.
  * [137] F Pepe, Stefano Cristiani, R Rebolo, NC Santos, H Dekker, A Cabral, Paolo Di Marcantonio, P Figueira, G Lo Curto, C Lovis, et al.  Espresso at vlt-on-sky performance and first results.  Astronomy & Astrophysics, 645:A96, 2021.
  * [138] Michael Perryman, Joel Hartman, Gáspár Á Bakos, and Lennart Lindegren.  Astrometric exoplanet detection with gaia.  The Astrophysical Journal, 797(1):14, 2014.
  * [139] RL Akeson, X Chen, D Ciardi, M Crane, J Good, M Harbut, E Jackson, SR Kane, AC Laity, S Leifer, et al.  The nasa exoplanet archive: data and tools for exoplanet research.  Publications of the Astronomical Society of the Pacific, 125(930):989, 2013.
  * [140] Wei Zhu and Subo Dong.  Exoplanet statistics and theoretical implications.  Annual Review of Astronomy and Astrophysics, 59:291–336, 2021.
  * [141] D. L. Pollacco, I. Skillen, A. Collier Cameron, D. J. Christian, C. Hellier, J. Irwin, T. A. Lister, R. A. Street, R. G. West, D. R. Anderson, W. I. Clarkson, H. Deeg, B. Enoch, A. Evans, A. Fitzsimmons, C. A. Haswell, S. Hodgkin, K. Horne, S. R. Kane, F. P. Keenan, P. F. L. Maxted, A. J. Norton, J. Osborne, N. R. Parley, R. S. I. Ryans, B. Smalley, P. J. Wheatley, and D. M. Wilson.  The WASP Project and the SuperWASP Cameras.  PASP, 118(848):1407–1418, October 2006.
  * [142] G. Bakos, R. W. Noyes, G. Kovács, K. Z. Stanek, D. D. Sasselov, and I. Domsa.  Wide-Field Millimagnitude Photometry with the HAT: A Tool for Extrasolar Planet Detection.  PASP, 116(817):266–277, March 2004.
  * [143] William J Borucki, David Koch, Gibor Basri, Natalie Batalha, Timothy Brown, Douglas Caldwell, John Caldwell, Jørgen Christensen-Dalsgaard, William D Cochran, Edna DeVore, et al.  Kepler planet-detection mission: introduction and first results.  Science, 327(5968):977–980, 2010.
  * [144] Debra A. Fischer, Guillem Anglada-Escude, Pamela Arriagada, Roman V. Baluev, Jacob L. Bean, Francois Bouchy, Lars A. Buchhave, Thorsten Carroll, Abhijit Chakraborty, Justin R. Crepp, Rebekah I. Dawson, Scott A. Diddams, Xavier Dumusque, Jason D. Eastman, Michael Endl, Pedro Figueira, Eric B. Ford, Daniel Foreman-Mackey, Paul Fournier, Gabor Fűrész, B. Scott Gaudi, Philip C. Gregory, Frank Grundahl, Artie P. Hatzes, Guillaume Hébrard, Enrique Herrero, David W. Hogg, Andrew W. Howard, John A. Johnson, Paul Jorden, Colby A. Jurgenson, David W. Latham, Greg Laughlin, Thomas J. Loredo, Christophe Lovis, Suvrath Mahadevan, Tyler M. McCracken, Francesco Pepe, Mario Perez, David F. Phillips, Peter P. Plavchan, Lisa Prato, Andreas Quirrenbach, Ansgar Reiners, Paul Robertson, Nuno C. Santos, David Sawyer, Damien Segransan, Alessandro Sozzetti, Tilo Steinmetz, Andrew Szentgyorgyi, Stéphane Udry, Jeff A. Valenti, Sharon X. Wang, Robert A. Wittenmyer, and Jason T. Wright.  State of the Field: Extreme Precision Radial Velocities.  PASP, 128(964):066001, June 2016.
  * [145] Brendan P. Bowler.  Imaging Extrasolar Giant Planets.  PASP, 128(968):102001, October 2016.
  * [146] Matthew T Penny, B Scott Gaudi, Eamonn Kerins, Nicholas J Rattenbury, Shude Mao, Annie C Robin, and Sebastiano Calchi Novati.  Predictions of the wfirst microlensing survey. i. bound planet detection rates.  The Astrophysical Journal Supplement Series, 241(1):3, 2019.
  * [147] Anna-Christina Eilers, David W Hogg, Hans-Walter Rix, and Melissa K Ness.  The circular velocity curve of the milky way from 5 to 25 kpc.  The Astrophysical Journal, 871(1):120, 2019.
  * [148] Konrad Kuijken and Gerard Gilmore.  The mass distribution in the galactic disc–ii. determination of the surface mass density of the galactic disc near the sun.  Monthly Notices of the Royal Astronomical Society, 239(2):605–649, 1989.
  * [149] Johan Holmberg and Chris Flynn.  The local density of matter mapped by hipparcos.  Monthly Notices of the Royal Astronomical Society, 313(2):209–216, 2000.
  * [150] Jo Bovy and Scott Tremaine.  On the local dark matter density.  The Astrophysical Journal, 756(1):89, 2012.
  * [151] Katelin Schutz, Tongyan Lin, Benjamin R Safdi, and Chih-Liang Wu.  Constraining a thin dark matter disk with g a i a.  Physical review letters, 121(8):081101, 2018.
  * [152] Axel Widmark.  Measuring the local matter density using gaia dr2.  Astronomy & Astrophysics, 623:A30, 2019.
  * [153] Morgan Bennett and Jo Bovy.  Vertical waves in the solar neighbourhood in gaia dr2.  Monthly Notices of the Royal Astronomical Society, 482(1):1417–1425, 2019.
  * [154] T Antoja, A Helmi, M Romero-Gómez, D Katz, C Babusiaux, Ronald Drimmel, DW Evans, F Figueras, Eloisa Poggio, C Reylé, et al.  A dynamically young and perturbed milky way disk.  Nature, 561(7723):360–362, 2018.
  * [155] Amina Helmi, Carine Babusiaux, Helmer H Koppelman, Davide Massari, Jovan Veljanoski, and Anthony GA Brown.  The merger that led to the formation of the milky way’s inner stellar halo and thick disk.  Nature, 563(7729):85–88, 2018.
  * [156] Aakash Ravi, Nicholas Langellier, David F Phillips, Malte Buschmann, Benjamin R Safdi, and Ronald L Walsworth.  Probing dark matter using precision measurements of stellar accelerations.  Physical Review Letters, 123(9):091101, 2019.
  * [157] Hamish Silverwood and Richard Easther.  Stellar accelerations and the galactic gravitational field.  Publications of the Astronomical Society of Australia, 36, 2019\.
  * [158] Sukanya Chakrabarti, Jason Wright, Philip Chang, Alice Quillen, Peter Craig, Joey Territo, Elena D’Onghia, Kathryn V Johnston, Robert J De Rosa, Daniel Huber, et al.  Toward a direct measure of the galactic acceleration.  The Astrophysical Journal Letters, 902(1):L28, 2020.
  * [159] David F Phillips, Aakash Ravi, Reza Ebadi, and Ronald L Walsworth.  Milky way accelerometry via millisecond pulsar timing.  Physical Review Letters, 126(14):141103, 2021.
  * [160] Sukanya Chakrabarti, Philip Chang, Michael T Lam, Sarah J Vigeland, and Alice C Quillen.  A measurement of the galactic plane mass density from binary pulsar accelerations.  The Astrophysical Journal Letters, 907(2):L26, 2021.
  * [161] Jo Bovy.  A purely acceleration-based measurement of the fundamental galactic parameters.  arXiv preprint arXiv:2012.02169, 2020.
  * [162] Lars Hernquist.  An Analytical Model for Spherical Galaxies and Bulges.  ApJ, 356:359, June 1990.
  * [163] Masanori Miyamoto and Ryuzaburo Nagai.  Three-dimensional models for the distribution of mass in galaxies.  Publications of the Astronomical Society of Japan, 27:533–543, 1975\.
  * [164] Julio F. Navarro, Carlos S. Frenk, and Simon D. M. White.  The Structure of Cold Dark Matter Halos.  ApJ, 462:563, May 1996.
  * [165] B. Paczynski.  Gravitational Microlensing by the Galactic Halo.  ApJ, 304:1, May 1986.
  * [166] E. Hog, I. D. Novikov, and A. G. Polnarev.  MACHO photometry and astrometry.  A&A, 294:287–294, February 1995.
  * [167] M. Miyamoto and Y. Yoshii.  Astrometry for Determining the MACHO Mass and Trajectory.  AJ, 110:1427, September 1995.
  * [168] Mark A. Walker.  Microlensed Image Motions.  ApJ, 453:37, November 1995.
  * [169] B. Paczynski.  Gravitational Microlensing of the Galactic Bulge Stars.  ApJ, 371:L63, April 1991.
  * [170] Kim Griest, Charles Alcock, Timothy S. Axelrod, David P. Bennett, Kem H. Cook, Kenneth C. Freeman, Hye-Sook Park, Saul Perlmutter, Bruce A. Peterson, Peter J. Quinn, Alexander W. Rodgers, Christopher W. Stubbs, and MACHO Collaboration.  Gravitational Microlensing as a Method of Detecting Disk Dark Matter and Faint Disk Stars.  ApJ, 372:L79, May 1991.
  * [171] Shude Mao and Bohdan Paczynski.  Gravitational Microlensing by Double Stars and Planetary Systems.  ApJ, 374:L37, June 1991.
  * [172] A. Udalski, M. Szymanski, J. Kaluzny, M. Kubiak, M. Mateo, W. Krzeminski, and B. Paczynski.  The Optical Gravitational Lensing Experiment. The Early Warning System: Real Time Microlensing.  Acta Astron., 44:227–234, July 1994.
  * [173] I. A. Bond, F. Abe, R. J. Dodd, J. B. Hearnshaw, M. Honda, J. Jugaku, P. M. Kilmartin, A. Marles, K. Masuda, Y. Matsubara, Y. Muraki, T. Nakamura, G. Nankivell, S. Noda, C. Noguchi, K. Ohnishi, N. J. Rattenbury, M. Reid, To. Saito, H. Sato, M. Sekiguchi, J. Skuljan, D. J. Sullivan, T. Sumi, M. Takeuti, Y. Watase, S. Wilkinson, R. Yamada, T. Yanagisawa, and P. C. M. Yock.  Real-time difference imaging analysis of MOA Galactic bulge observations during 2000.  MNRAS, 327(3):868–880, November 2001.
  * [174] Seung-Lee Kim, Chung-Uk Lee, Byeong-Gon Park, Dong-Jin Kim, Sang-Mok Cha, Yongseok Lee, Cheongho Han, Moo-Young Chun, and Insoo Yuk.  KMTNET: A Network of 1.6 m Wide-Field Optical Telescopes Installed at Three Southern Observatories.  Journal of Korean Astronomical Society, 49(1):37–44, February 2016\.
  * [175] D. Spergel, N. Gehrels, C. Baltay, D. Bennett, J. Breckinridge, M. Donahue, A. Dressler, B. S. Gaudi, T. Greene, O. Guyon, C. Hirata, J. Kalirai, N. J. Kasdin, B. Macintosh, W. Moos, S. Perlmutter, M. Postman, B. Rauscher, J. Rhodes, Y. Wang, D. Weinberg, D. Benford, M. Hudson, W. S. Jeong, Y. Mellier, W. Traub, T. Yamada, P. Capak, J. Colbert, D. Masters, M. Penny, D. Savransky, D. Stern, N. Zimmerman, R. Barry, L. Bartusek, K. Carpenter, E. Cheng, D. Content, F. Dekens, R. Demers, K. Grady, C. Jackson, G. Kuan, J. Kruk, M. Melton, B. Nemati, B. Parvin, I. Poberezhskiy, C. Peddie, J. Ruffa, J. K. Wallace, A. Whipple, E. Wollack, and F. Zhao.  Wide-Field InfrarRed Survey Telescope-Astrophysics Focused Telescope Assets WFIRST-AFTA 2015 Report.  arXiv e-prints, page arXiv:1503.03757, March 2015.
  * [176] J. R. Lu, E. Sinukoff, E. O. Ofek, A. Udalski, and S. Kozlowski.  A Search For Stellar-mass Black Holes Via Astrometric Microlensing.  ApJ, 830(1):41, October 2016.
  * [177] N. Kains, A. Calamida, K. C. Sahu, S. Casertano, J. Anderson, A. Udalski, M. Zoccali, H. Bond, M. Albrow, I. Bond, T. Brown, M. Dominik, C. Fryer, M. Livio, S. Mao, and M. Rejkuba.  Microlensing Constraints on the Mass of Single Stars from HST Astrometric Measurements.  ApJ, 843(2):145, July 2017.
  * [178] Krzysztof A. Rybicki, Łukasz Wyrzykowski, Jakub Klencki, Jos de Bruijne, Krzysztof Belczyński, and Martyna Chruślińska.  On the accuracy of mass measurement for microlensing black holes as seen by Gaia and OGLE.  MNRAS, 476(2):2013–2028, May 2018.
  * [179] Kailash C. Sahu, Jay Anderson, Stefano Casertano, Howard E. Bond, Pierre Bergeron, Edmund P. Nelan, Laurent Pueyo, Thomas M. Brown, Andrea Bellini, Zoltan G. Levay, Joshua Sokol, Martin Dominik, Annalisa Calamida, Noé Kains, and Mario Livio.  Relativistic deflection of background starlight measures the mass of a nearby white dwarf star.  Science, 356(6342):1046–1050, June 2017.
  * [180] A. Zurlo, R. Gratton, D. Mesa, S. Desidera, A. Enia, K. Sahu, J. M. Almenara, P. Kervella, H. Avenhaus, J. Girard, M. Janson, E. Lagadec, M. Langlois, J. Milli, C. Perrot, J. E. Schlieder, C. Thalmann, A. Vigan, E. Giro, L. Gluck, J. Ramos, and A. Roux.  The gravitational mass of Proxima Centauri measured with SPHERE from a microlensing event.  MNRAS, 480(1):236–244, October 2018.
  * [181] Casey Y. Lam, Jessica R. Lu, Andrzej Udalski, Ian Bond, David P. Bennett, Jan Skowron, Przemek Mróz, Radek Poleski, Takahiro Sumi, Michał K. Szymański, Szymon Kozłowski, Paweł Pietrukowicz, Igor Soszyński, Krzysztof Ulaczyk, Łukasz Wyrzykowski, Shota Miyazaki, Daisuke Suzuki, Naoki Koshimoto, Nicholas J. Rattenbury, Matthew W. Hosek, Fumio Abe, Richard Barry, Aparna Bhattacharya, Akihiko Fukui, Hirosane Fujii, Yuki Hirao, Yoshitaka Itow, Rintaro Kirikawa, Iona Kondo, Yutaka Matsubara, Sho Matsumoto, Yasushi Muraki, Greg Olmschenk, Clément Ranc, Arisa Okamura, Yuki Satoh, Stela Ishitani Silva, Taiga Toda, Paul J. Tristram, Aikaterini Vandorou, Hibiki Yama, Natasha S. Abrams, Shrihan Agarwal, Sam Rose, and Sean K. Terry.  Supplement: “An Isolated Mass-gap Black Hole or Neutron Star Detected with Astrometric Microlensing” (2022, ApJL, 933, L23).  ApJS, 260(2):55, June 2022.
  * [182] Kailash C. Sahu, Jay Anderson, Stefano Casertano, Howard E. Bond, Andrzej Udalski, Martin Dominik, Annalisa Calamida, Andrea Bellini, Thomas M. Brown, Marina Rejkuba, Varun Bajaj, Noé Kains, Henry C. Ferguson, Chris L. Fryer, Philip Yock, Przemek Mróz, Szymon Kozłowski, Paweł Pietrukowicz, Radek Poleski, Jan Skowron, Igor Soszyński, Michał K. Szymański, Krzysztof Ulaczyk, Łukasz Wyrzykowski, Richard K. Barry, David P. Bennett, Ian A. Bond, Yuki Hirao, Stela Ishitani Silva, Iona Kondo, Naoki Koshimoto, Clément Ranc, Nicholas J. Rattenbury, Takahiro Sumi, Daisuke Suzuki, Paul J. Tristram, Aikaterini Vandorou, Jean-Philippe Beaulieu, Jean-Baptiste Marquette, Andrew Cole, Pascal Fouqué, Kym Hill, Stefan Dieters, Christian Coutures, Dijana Dominis-Prester, Clara Bennett, Etienne Bachelet, John Menzies, Michael Albrow, Karen Pollard, Andrew Gould, Jennifer C. Yee, William Allen, Leonardo A. Almeida, Grant Christie, John Drummond, Avishay Gal-Yam, Evgeny Gorbikov, Francisco Jablonski, Chung-Uk Lee, Dan Maoz, Ilan Manulis, Jennie McCormick, Tim Natusch, Richard W. Pogge, Yossi Shvartzvald, Uffe G. Jørgensen, Khalid A. Alsubai, Michael I. Andersen, Valerio Bozza, Sebastiano Calchi Novati, Martin Burgdorf, Tobias C. Hinse, Markus Hundertmark, Tim-Oliver Husser, Eamonn Kerins, Penelope Longa-Peña, Luigi Mancini, Matthew Penny, Sohrab Rahvar, Davide Ricci, Sedighe Sajadian, Jesper Skottfelt, Colin Snodgrass, John Southworth, Jeremy Tregloan-Reed, Joachim Wambsganss, Olivier Wertz, Yiannis Tsapras, Rachel A. Street, D. M. Bramich, Keith Horne, Iain A. Steele, and RoboNet Collaboration.  An Isolated Stellar-mass Black Hole Detected through Astrometric Microlensing.  ApJ, 933(1):83, July 2022.
  * [183] Klüter, J., Bastian, U., and Wambsganss, J.  Expectations on mass determination using astrometric microlensing by gaia.  A&A, 640:A83, 2020.
  * [184] I-Kai Chen, Marius Kongsore, and Ken Van Tilburg.  Detecting Dark Compact Objects in Gaia DR4: A Data Analysis Pipeline for Transient Astrometric Lensing Searches.  arXiv e-prints, page arXiv:2301.00822, January 2023.
  * [185] F. Delplancke, K. M. Górski, and A. Richichi.  Resolving gravitational microlensing events with long-baseline optical interferometry. Prospects for the ESO Very Large Telescope Interferometer.  A&A, 375:701–710, August 2001.
  * [186] Klüter, J., Bastian, U., Demleitner, M., and Wambsganss, J.  Prediction of astrometric microlensing events from gaia dr2 proper motions.  A&A, 620:A175, 2018.
  * [187] Leonidas Moustakas, Matthew O’Dowd, Timo Anguita, Rachel Webster, George Chartas, Matthew Cornachione, Xinyu Dai, Carina Fian, Damien Hutsemekers, Jorge Jimenez-Vicente, et al.  Astro2020 science white paper-quasar microlensing: Revolutionizing our understanding of quasar structure and dynamics.  arXiv preprint arXiv:1904.12967, 2019.
  * [188] Marie Treyer and Joachim Wambsganss.  Astrometric microlensing of quasars-dependence on surface mass density and external shear.  Astronomy & Astrophysics, 416(1):19–34, 2004.
  * [189] Walsh Dennis, Robert F Carswell, and Ray J Weymann.  0957+561 a, b: twin quasistellar objects or gravitational lens?  Nature, 279(5712):381–384, 1979.
  * [190] John Huchra, Marc Gorenstein, Stephen Kent, Irwin Shapiro, Graeme Smith, Edward Horine, and Richard Perley.  2237+ 0305: A new and unusual gravitational lens.  The Astronomical Journal, 90:691–696, 1985.
  * [191] CS Kochanek.  Quantitative interpretation of quasar microlensing light curves.  The Astrophysical Journal, 605(1):58, 2004.
  * [192] Naohisa Inada, Masamune Oguri, Robert H Becker, Min-Su Shin, Gordon T Richards, Joseph F Hennawi, Richard L White, Bartosz Pindor, Michael A Strauss, Christopher S Kochanek, et al.  The sloan digital sky survey quasar lens search. ii. statistical lens sample from the third data release.  The Astronomical Journal, 135(2):496, 2008.
  * [193] Ana M Mosquera and Christopher S Kochanek.  The microlensing properties of a sample of 87 lensed quasars.  The Astrophysical Journal, 738(1):96, 2011.
  * [194] Masamune Oguri and Philip J Marshall.  Gravitationally lensed quasars and supernovae in future wide-field optical imaging surveys.  Monthly Notices of the Royal Astronomical Society, 405(4):2579–2593, 2010.
  * [195] M Jaroszynski, J Wambsganss, and B Paczynski.  Microlensed light curves for thin accretion disks around schwarzschild and kerr black holes.  The Astrophysical Journal, 396:L65–L68, 1992.
  * [196] Kevin P Rauch and Roger D Blandford.  Microlensing and the structure of active galactic nucleus accretion disks.  The Astrophysical Journal, 381:L39–L42, 1991.
  * [197] Joachim Wambsganß, B Paczynski, and Peter Schneider.  Interpretation of the microlensing event in qso 2237+ 0305.  The Astrophysical Journal, 358:L33–L36, 1990.
  * [198] RL Webster, AMN Ferguson, RT Corrigan, and MJ Irwin.  Interpreting the light curve of q2237+ 0305.  The Astronomical Journal, 102:1939–1945, 1991.
  * [199] L. L. R. Williams and P. Saha.  Improper Motions in Lensed QSOs.  AJ, 110:1471, October 1995.
  * [200] Geraint F Lewis and Rodrigo A Ibata.  Quasar image shifts resulting from gravitational microlensing.  The Astrophysical Journal, 501(2):478, 1998.
  * [201] R Kayser, S Refsdal, and R Stabell.  Astrophysical applications of gravitational micro-lensing.  Astronomy and Astrophysics, 166:36–52, 1986.
  * [202] Xiaohui Fan, Feige Wang, Jinyi Yang, Charles R Keeton, Minghao Yue, Ann Zabludoff, Fuyan Bian, Marco Bonaglia, Iskren Y Georgiev, Joseph F Hennawi, et al.  The discovery of a gravitationally lensed quasar at z = 6.51.  The Astrophysical Journal Letters, 870(2):L11, 2019.
  * [203] A. M. Ghez, B. L. Klein, M. Morris, and E. E. Becklin.  High proper-motion stars in the vicinity of sagittarius a*: Evidence for a supermassive black hole at the center of our galaxy.  The Astrophysical Journal, 509(2):678, dec 1998.
  * [204] A. M. Ghez, S. Salim, N. N. Weinberg, J. R. Lu, T. Do, J. K. Dunn, K. Matthews, M. R. Morris, S. Yelda, E. E. Becklin, T. Kremenek, M. Milosavljevic, and J. Naiman.  Measuring Distance and Properties of the Milky Way’s Central Supermassive Black Hole with Stellar Orbits.  ApJ, 689(2):1044–1062, December 2008.
  * [205] S. Gillessen, F. Eisenhauer, S. Trippe, T. Alexander, R. Genzel, F. Martins, and T. Ott.  Monitoring Stellar Orbits Around the Massive Black Hole in the Galactic Center.  ApJ, 692(2):1075–1109, February 2009.
  * [206] Marios Galanis et al.  In preparation, 2023.
  * [207] Kip S. Thorne and Roger D. Blandford.  Modern Classical Physics.  Princeton University Press, Princeton, NJ, 2017.