# Extended-Path Intensity Correlation: Microarcsecond Astrometry with an
Arcsecond Field of View

Marios Galanis  Ken Van Tilburg  Masha Baryakhtar  and Neal Weiner

###### Abstract

We develop in detail a recently proposed optical-path modification of
astronomical intensity interferometers. Extended-Path Intensity Correlation
(EPIC) [1] introduces a tunable path extension, enabling differential
astrometry of multiple compact sources such as stars and quasars at
separations of up to a few arcseconds. Combined with other recent
technological advances in spectroscopy and fast single-photon detection, a
ground-based intensity interferometer array can achieve microarcsecond
resolution and even better light-centroiding accuracy on bright sources of
magnitude $m\lesssim 15$. We lay out the theory and technical requirements of
EPIC, and discuss the scientific potential. Promising applications include
astrometric lensing of stars and quasar images, binary-orbit characterization,
exoplanet detection, Galactic acceleration measurements and calibration of the
cosmic distance ladder. The introduction of the path extension thus
significantly increases the scope of intensity interferometry while reaching
unprecedented levels of relative astrometric precision.

##  1 Introduction

Astrometry is the oldest science: the motions of celestial objects across the
sky have been used for millennia to understand the cosmos and our place in it.
In the present day, precision measurements of stellar velocities in our Galaxy
and Local Group have provided invaluable information on galactic dynamics and
structure formation. Relative velocities of stars in the central region of the
Milky Way unveiled the supermassive black hole Sagittarius A*, while stellar
parallax established the first rungs of the distance ladder used to discover
the accelerated expansion of the Universe. In recent years, advances in
astrometry have made possible the identification of isolated black holes [2,
3, 4] and exoplanets [5], discoveries of new stellar substructures [6, 7, 8],
a clearer picture of the formation and evolution of the Milky Way [9],
stringent tests of gravity [10, 11, 12, 13], and searches for time-domain
lensing by small dark matter halos and compact objects [14], to name but a few
applications. Future measurements may reveal the gravitational potential of
the Milky Way [15], constraints on dark matter substructure [16], and low-
frequency gravitational waves [17, 18, 19, 20]. These applications are enabled
by the precision astrometry of space-based telescopes such as Hipparcos [21],
Gaia [22], Hubble Space Telescope (HST) [23], and James Webb Space Telescope
(JWST) [24], and of ground-based amplitude interferometers in the optical and
radio bands such as GRAVITY at VLT [25], the Center for High Angular
Resolution Astronomy (CHARA) [26], and the Event Horizon Telescope (EHT) [27,
28], which have reached light-centroiding precisions on the order of tens of
microarcseconds.

These observatories are all contending with a practical obstacle: diffraction-
limited resolution, approximately equal to the wavelength of light divided by
the observatory’s effective size. For imaging telescopes, this size is the
aperture diameter, whereas for interferometers it is the baseline distance:
the separation between telescopes or receivers. Imaging telescopes can achieve
a precision on the light centroid of a point source that far surpasses the
angular resolution limit, as long as the exposure is sufficiently long, the
point spread function (PSF) is stable, and other systematics are kept under
control. In very long baseline interferometry (VLBI) in the radio band (up to
millimeter wavelengths), receivers record the amplitude and phase of the
electromagnetic field at different corners of the globe [28]. The data is then
correlated offline, yielding information at an angular scale of tens of
microarcseconds. The size of ground-based VLBI is already maximized, while
shorter wavelengths present their own difficulties. Astronomical
interferometers in the infrared and optical bands have to physically recombine
the light from separate telescopes (since receiver electronics cannot keep
track of the rapidly oscillating phase) and keep their separations stable to a
fraction of wavelength, limiting their baselines to hundreds of meters at most
[29, 26].

Intensity interferometry combines the best of both worlds—short wavelengths
yet long baselines (with milder tolerance) and off-line correlation. The
technique relies on second-order coherence: interference of the intensities,
rather than the amplitudes, of incoming light, so only the photon arrival
times have to be recorded and correlated. It was conceived theoretically by
Hanbury Brown and Twiss in 1954 [30], who only two years later demonstrated
the effect experimentally in the laboratory, using light from a mercury arc
lamp [31]. Later in 1956, with a rudimentary setup and poor viewing
conditions, Hanbury Brown used intensity interferometry to provide the first
measurement of the $0.0063\,\mathrm{arcsec}$ angular diameter of Sirius [32],
a spectacular initiation of a new method in observational astronomy. The
program was then extended [33, 34, 35, 36], and culminated with the Narrabri
Stellar Intensity Interferometer (NSII) [37, 38, 39]. In the mid-1970s, the
NSII reported the angular diameters of 32 bright stars down to
$30\,\mathrm{\mu as}$ precision [40, 41], as well as orbital parameters,
masses, and line-of-sight distance of a close stellar binary system [42]. The
advent of ever larger arrays of Imaging Atmospheric Cherenkov Telescopes
(IACTs) [43, 44, 45, 46, 47] to detect high-energy gamma rays has spurred a
recent revival of intensity interferometry [48, 49, 50, 51, 52, 53, 54, 55,
56, 57]; in addition new fast photodetector technology has enabled the
development of smaller telescope arrays with competitive sensitivity [58].

The advantages of intensity interferometry are unfortunately paired with two
significant drawbacks. The first is the signal-to-noise ratio (SNR), as
intensity correlations only receive contributions from “nearly coincident”
photons in at least two receivers. Since the intensity of starlight typically
fluctuates on femtosecond time scales, the much longer timing resolution of
photon counters smears out the intensity correlations and reduces the fringe
contrast; the NSII measurements were limited to very bright stars of magnitude
below 3. The second limitation of traditional intensity interferometry is the
extremely narrow field of view (FOV), which is naturally of the same
parametric size as the angular resolution itself. The combination of the small
SNR and FOV have limited astronomical intensity interferometry almost
exclusively to studies of the emission morphology of isolated bright stars,
and a few very closely separated multiple-star systems [42, 59].

In this work, we detail the _Extended-Path Intensity Correlation_ (EPIC)
observatory design which addresses both the low SNR and small FOV drawbacks of
intensity interferometry [1]; a schematic of the design is shown in figure 1.
We outline how recent technological advances in multichannel spectroscopic
readout and fast single photon detection open up intensity interferometry to
much fainter sources. The advantages of a multichannel spectroscopic readout
were already known by Hanbury Brown [39], and its pairing with modern
photodetectors was laid out clearly in ref. [50], which argued that one could
achieve limiting R-band magnitudes of about 14. Spectroscopic splitting along
with the introduction of beamsplitters to improve the sensitivity and
information gathered from the telescopes was also proposed in [60, 61]. Here,
we summarize further recent improvements in photodetection technology and
provide a detailed quantitative description of a multichannel spectroscopic
readout in the context of intensity interferometry.

Figure 1: Diagram of an extended-path intensity correlator (EPIC). Two sources
$i=a,b$ separated by an angle
$\boldsymbol{\mathbf{\theta}}_{ba}=\hat{\boldsymbol{\mathbf{\theta}}}_{b}-\hat{\boldsymbol{\mathbf{\theta}}}_{a}$
are simultaneously observed by two telescopes $p=1,2$; the respective
distances are labeled as $\boldsymbol{\mathbf{r}}_{ip}$. In each telescope,
the light from both sources is collected and collimated into the “path
extension” stage, where a beamsplitter directs the light along two paths, one
longer than the other by a variable amount $\ell_{p}$, and a second
beamsplitter recombines the beam. The light is then split into multiple
subapertures, and a dispersive element spectrally splits the light. Ultrafast
single-photon detectors record the arrival times of photons in each spectral
channel, which are ultimately correlated offline after applying a time offset
$\tau$. The intensity correlations provide information on the separation
$\boldsymbol{\mathbf{\theta}}_{ba}$ of the two sources as well as their
angular sizes.

To overcome the very narrow field of view, EPIC uses a tunable hardware
modification to the constituent telescopes of an astronomical intensity
interferometer array. This setup requires a pair of adjustable beamsplitters
and mirrors that can add a differential path extension to the incoming light,
thus superposing intensity fluctuations from sources at different locations on
the celestial sphere [1].111A related technique called “time-delay
interferometry” has been proposed previously in the context of gravitational
radiation sensing with (amplitude) interferometric detectors of unequal arm
lengths, where a path extension (i.e. geometric time delay) is introduced to
eliminate laser phase noise [62]. Likewise, ref. [63] proposed to test quantum
mechanics in photon coincidence experiments with a logically similar optical
setup. The resulting variable path extension (and wavefront correction)
dramatically widens the FOV of ground-based intensity interferometry, up to a
few arcseconds, the isoplanatic angle of the atmosphere. This alteration would
drastically increase the scope of intensity interferometry, enabling
differential astrometry at an unprecedented precision and accuracy for a wide
range of compact celestial light sources.

In section 2, we review the basic theory of intensity interferometry (section
2.1), describe the addition of the optical path extension (section 2.2),
discuss typical celestial sources of finite angular extent (section 2.3),
calculate the idealized signal-to-noise ratio and sensitivity (section 2.4),
and calculate field of view limitations from atmospheric aberrations (section
2.5). We outline the technical requirements in section 3, including telescope
design and array layout (section 3.1), observational procedure (section 3.2),
photodetection (section 3.3), dispersive elements for spectroscopy (section
3.4), and optical tolerances (section 3.5). We propose a sample of scientific
applications with EPIC in section 4: binary orbits (section 4.1), exoplanets
(section 4.2), Galactic accelerations (section 4.3), stellar microlensing
(section 4.4), and strongly lensed quasars (section 4.5). We conclude in
section 5. The appendices contain a list of the symbols used throughout the
text (appendix A), details of the derivations summarized in the text (appendix
B), and a wave-optics treatment of our setup’s optical elements (appendix C).

##  2 Theory

The basic intensity interferometry setup consists of at least two telescopes
collecting photons—and precisely recording their arrival times—from a small
region of the sky. This is to be contrasted with _amplitude_ interferometers,
which record the electric field amplitudes as a function of time at a pair of
receivers separated by a baseline $\boldsymbol{\mathbf{d}}$. The two-point
correlation function of electric field amplitudes is proportional to the
“complex visibility”: the Fourier transform of the time-averaged intensity
distribution of the image at angular wavenumber
$\overline{k}\boldsymbol{\mathbf{d}}_{\perp}$ conventionally decomposed into
$(u,v)$ coordinates, where $\overline{k}$ is the mean wavenumber of the
electromagnetic wave, and $\boldsymbol{\mathbf{d}}_{\perp}$ is the baseline
vector projection perpendicular to the line of sight. Through photon
collection, an _intensity_ interferometer samples intensities (the square of
the field amplitudes) at separate locations and compares their fluctuations.
The excess correlation of the intensities $C(\boldsymbol{\mathbf{d}},\tau)$ at
points separated by distance $\boldsymbol{\mathbf{d}}$ and time $\tau$ is
proportional to a specific four-point function of the electromagnetic field:
the square of the modulus of the complex visibility at angular wavenumber
$\overline{k}\boldsymbol{\mathbf{d}}_{\perp}$. The overall phase information
is lost, which drastically degrades _global_ astrometric precision, but
information about relative phases is retained, thus providing excellent
_differential_ astrometry.

Figure 2: Left: Same as figure 1, but with the two-photon amplitude
corresponding to the fringe in eqs. 2.25 and 2.26 color coded in the path
extension stage: orange corresponds to the amplitude from source $a$, blue
from source $b$. The path extensions are tuned such that
$\ell_{1}+\ell_{2}=(r_{a2}+r_{b1})-(r_{a1}+r_{b2})=\boldsymbol{\mathbf{\theta}}_{ba}\cdot\boldsymbol{\mathbf{d}}$.
Right: Excess fractional intensity correlation
$C(\boldsymbol{\mathbf{d}},\tau)$ as a function timing offset $\tau$ for a
single point-like source at celestial position
$\hat{\boldsymbol{\mathbf{\theta}}}_{a}$, for timing resolution
$\sigma_{t}=30\,\text{ps}$, and central wavenumber
$\overline{k}=2\pi/500\,\text{nm}$ and spectral resolution
$\mathcal{R}\equiv\overline{k}/\sigma_{k}=5{,}000$. The black curve shows the
intensity fringe of eq. 2.11 for an intensity interferometer without path
extension, while the orange curve shows the four fringes of eq. 2.23 for an
_extended-path_ intensity correlator. The path extension retains the original
main fringe at 1/4 the contrast, but also produces three equally bright ghost
fringes, as if there were 3 other images located at
$\hat{\boldsymbol{\mathbf{\theta}}}_{a}+\frac{\ell_{1}}{d}\hat{\boldsymbol{\mathbf{d}}}$,
$\hat{\boldsymbol{\mathbf{\theta}}}_{a}-\frac{\ell_{2}}{d}\hat{\boldsymbol{\mathbf{d}}}$,
$\hat{\boldsymbol{\mathbf{\theta}}}_{a}-\frac{\ell_{2}-\ell_{1}}{d}\hat{\boldsymbol{\mathbf{d}}}$,
respectively. The fringe corresponding to the paths from source $a$ shown in
the left panel is the leftmost fringe highlighted in bold.

Our main new idea is to enlarge the scope (literally and figuratively) of
intensity interferometry by introducing a variable, probabilistic path
extension inside each constituent telescope of the intensity interferometer
array while keeping the same angular resolution and light-centroiding
precision. This is done by introducing additional adjustable optical elements
as shown schematically in figure 2 (left). In a conventional intensity
interferometer, the excess intensity correlation between a pair of telescopes
for a point-like source at sky location
$\hat{\boldsymbol{\mathbf{\theta}}}_{a}$ occurs near a time offset of
$\tau=-\boldsymbol{\mathbf{d}}\cdot\hat{\boldsymbol{\mathbf{\theta}}}_{a}/c$
with $c$ the speed of light, depicted in figure 2 (right) by the black line.
In an Extended-Path Intensity Correlator (EPIC), the path extension stage at
telescope $p$ splits the beam into two, one geometrically longer than the
other by an amount $\ell_{p}$. The intensity correlation fringe is thus split
into $2^{2}$ fringes, each of $1/4$ the fringe contrast, depicted as orange
lines in figure 2 (right): the “original” fringe at
$c\tau=-\boldsymbol{\mathbf{d}}\cdot\hat{\boldsymbol{\mathbf{\theta}}}_{a}$,
one only delayed in telescope 1 at
$c\tau=-\boldsymbol{\mathbf{d}}\cdot\hat{\boldsymbol{\mathbf{\theta}}}_{a}-\ell_{1}$,
one only delayed in telescope 2 at
$c\tau=-\boldsymbol{\mathbf{d}}\cdot\hat{\boldsymbol{\mathbf{\theta}}}_{a}+\ell_{2}$,
and one delayed in both at
$c\tau=-\boldsymbol{\mathbf{d}}\cdot\hat{\boldsymbol{\mathbf{\theta}}}_{a}+\ell_{2}-\ell_{1}$.
This procedure can be used to “superpose” the intensity correlation fringes of
two sources $a$ and $b$, at locations $\hat{\boldsymbol{\mathbf{\theta}}}_{a}$
and $\hat{\boldsymbol{\mathbf{\theta}}}_{b}$, to measure their relative
angular separation
$\boldsymbol{\mathbf{\theta}}_{ba}\equiv\hat{\boldsymbol{\mathbf{\theta}}}_{b}-\hat{\boldsymbol{\mathbf{\theta}}}_{a}$
with extreme precision. For example, the intensity correlation fringe of
source $a$ delayed by $\ell_{1}$ in telescope 1, and that of source $b$
delayed by $\ell_{2}$ in telescope 2, will coincide (and yield mutual
intensity interference) when:

$\displaystyle\boxed{\ell_{1}+\ell_{2}=\boldsymbol{\mathbf{\theta}}_{ba}\cdot\boldsymbol{\mathbf{d}}.}$
(2.1)

Other permutations of fringes are also possible. In this way, the maximum
angular separation for intensity interferometry can be enlarged up to the
$\mathcal{O}(\mathrm{arcsec})$ isoplanatic angle determined by atmospheric
aberrations while retaining exquisite angular resolution. Hence, the dynamic
range (i.e. angular separation divided by resolution) of EPIC is many orders
of magnitude larger than that of traditional intensity interferometry.

In section 2.1, we present a classical description of intensity interferometry
in idealized scenarios, based on but significantly extending the review of
ref. [64]. More sophisticated treatments, including a quantum description and
the first- and second-order coherence properties of light, can be found e.g.
in ref. [65]. For a detailed review on optical stellar interferometry, see
e.g. ref. [66]; for recent developments see e.g. refs. [67, 68]. In section
2.2, we describe the path extension theory of EPIC. We review basic properties
of the most relevant astrophysical sources in section 2.3 in order to estimate
the signal-to-noise ratio and light-centroiding precision in section 2.4. We
calculate the atmospheric limitations on EPIC in section 2.5.

###  2.1 Intensity interferometry review

####  2.1.1 General

Suppose we have a single-photon detector recording a time series of photon
counts impinging on a telescope. We will assume the light in the telescope is
a superposition of a very large number $N_{\theta}$ of discrete emitters $i$,
each emitting a large number $N_{k}$ of modes $\alpha$ with random phases
$\phi^{\text{em}}_{i\alpha}$, giving an electric field at detector $p$,

$\displaystyle
E_{p}(t_{p})\equiv\sum_{i=1}^{N_{\theta}}\sum_{\alpha=1}^{N_{k}}E_{i\alpha}\exp\left\\{i\left[k_{\alpha}\left(ct_{p}-r_{ip}\right)+\phi^{\text{em}}_{i\alpha}+\widetilde{\phi}^{(p)}_{i\alpha}(t_{p})\right]\right\\}\equiv\sum_{i,\alpha}\widetilde{E}_{i\alpha}^{(p)}$
(2.2)

at a time $t_{p}$ kept by the clock associated with each photodetector $p$. In
the above, $k_{\alpha}$ is the wavenumber of each mode $\alpha$;
$r_{ip}=|\boldsymbol{\mathbf{r}}_{ip}|$ is the length of the vector
$\boldsymbol{\mathbf{r}}_{ip}\equiv\boldsymbol{\mathbf{r}}_{i}-\boldsymbol{\mathbf{r}}_{p}$
pointing from the photodetector $p$ to the emitter $i$. Additional phases that
may accumulate along the line of sight due to atmospheric fluctuations and
telescope imperfections are denoted by
$\widetilde{\phi}^{(p)}_{i\alpha}(t_{p})$ but will be suppressed until section
2.5 and section 3.5. As will be discussed in sections 2.5 and 3, these path-
dependent phases will ultimately limit the field of view of the intensity
interferometry setup and place constraints on the telescope quality. Without
loss of generality, the field amplitude $E_{i\alpha}$ associated with each
mode can be taken to be real; we defined the complex amplitudes
$\widetilde{E}_{i\alpha}^{(p)}$ in the last part of eq. 2.2 to simplify
subsequent notation. Finally, we treat the field amplitudes as scalars for
clarity of exposition; for unpolarized sources, intensity correlations signals
are reduced by a factor of $1/2$, which will be accounted for in section 2.4.

Throughout this work, we will assume that the detector separations are much
smaller than the (transversely projected) separations of the sources, which
are in turn taken to be much smaller than the line-of-sight distances (small
angle approximation). This implies that the angular separation of two sources
as observed from vantage point $p$,

$\displaystyle\boldsymbol{\mathbf{\theta}}_{ij}\equiv\hat{\boldsymbol{\mathbf{r}}}_{ip}-\hat{\boldsymbol{\mathbf{r}}}_{jp}\simeq\hat{\boldsymbol{\mathbf{\theta}}}_{i}-\hat{\boldsymbol{\mathbf{\theta}}}_{j},$
(2.3)

is independent of $p$, and that $|\boldsymbol{\mathbf{\theta}}_{ij}|\ll 1$.
Note that this also implies that electric field magnitude $E_{i\alpha}$ is
independent of $p$ for each mode. See figure 1 for the basic geometry in the
case of just two sources $i=a,b$.

The photodetectors record intensities
$I_{p}(t_{p})=\frac{1}{2}|E_{p}(t_{p})|^{2}$, with expected values:

$\displaystyle\left<I\right>\equiv\left<I_{p}(t_{p})\right>_{\phi^{\text{em}}}=\frac{1}{2}\sum_{i,\alpha}\left(E_{i\alpha}\right)^{2}\rightarrow\int\text{d}k\int\text{d}\Omega\,\frac{\text{d}I}{\text{d}k\,\text{d}\Omega},$
(2.4)

where the average is taken over the emitter phases
$\phi^{\text{em}}_{i\alpha}$, leaving only the “diagonal” terms. On the right,
we have taken the continuum limit, i.e. $N_{\theta},N_{k}\to\infty$, giving an
integral over wavenumbers $k$ and solid angle $\Omega$ over the field of view
of the telescope. The expected differential intensity
$\frac{\text{d}I}{\text{d}k\,\text{d}\Omega}$ is to be understood as a
convolution of the telescope-photodetector response with the source’s
intrinsic spectrum and image intensity. With our approximations and
assumptions of identical photodetectors and telescopes, the intensities are
independent of $p$. An illustration of the intensity impinging on a detector
with finite time resolution is given in figure 3.

Figure 3:  First row: Instantaneous intensity $I_{p}$ normalized to the true
mean intensity $\langle I_{p}\rangle$ from a single source as observed by two
telescopes labeled $p=1,2$ (left, right), for light with a Gaussian spectrum
with mean wavenumber $\overline{k}=2\pi/500\,\mathrm{nm}$ and standard
deviation $\sigma_{k}=\overline{k}/5000$. The true instantaneous intensity has
order-unity fractional fluctuations, a coherence time of order
$1/c\sigma_{k}$, and is _identical_ in both telescopes up to a time shift of
$t_{2}-t_{1}=50\,\mathrm{ps}$ due to the source’s wavefronts arriving earlier
in telescope 1. Second row: Photon arrival times (red sticks) in telescope 1
(left) and 2 (right) as a function of time. The photon detection probability
is directly proportional to the instantaneous intensity from the top panel(s).
We plot in blue the estimated intensities $\hat{I}_{p}$ (eq. 2.43) divided by
their estimated time averages $\overline{\hat{I}_{p}}$ (cfr. eq. 2.45),
assuming each telescope has a timing resolution of $\sigma_{t}/\sqrt{2}$ with
$\sigma_{t}=10\,\mathrm{ps}$. Third Row: Reconstructed excess fractional
intensity correlation $\hat{C}$ (green line, eq. 2.44) as a function of timing
offset $\tau=t_{2}-t_{1}$ based on the estimated intensities from the middle
panels with an observation time of $t_{\mathrm{obs}}=10^{3}\,\mathrm{ps}$;
green and yellow bands indicate $68\%$ and $95\%$ CL intervals. The true
excess correlation signal $C=\langle I_{1}I_{2}\rangle/\langle
I_{1}\rangle\langle I_{2}\rangle-1$ (black) is too small to be detected.
Fourth row: Same as in the third row, but for a much longer total observation
time $t_{\mathrm{obs}}=10^{6}\,\mathrm{ps}$, whereafter the excess correlation
at $\tau=50\,\mathrm{ps}$ is detected at high significance.

The intensity correlation between two detectors $p=1,2$ is:

$\displaystyle\left<I_{1}(t_{1})I_{2}(t_{2})\right>_{\phi^{\text{em}}}$
$\displaystyle=\frac{1}{4}\sum_{\begin{subarray}{c}i_{1},j_{1}\\\
i_{2},j_{2}\end{subarray}}\sum_{\begin{subarray}{c}\alpha_{1},\beta_{1}\\\
\alpha_{2},\beta_{2}\end{subarray}}\widetilde{E}^{(1)}_{i_{1}\alpha_{1}}\widetilde{E}^{(1)*}_{j_{1}\beta_{1}}\widetilde{E}^{(2)}_{i_{2}\alpha_{2}}\widetilde{E}_{j_{2}\beta_{2}}^{(2)*}$
$\displaystyle=\left(\frac{1}{2}\sum_{i,\alpha}\left(E_{i\alpha}\right)^{2}\right)^{2}+\left|\frac{1}{2}\sum_{i,\alpha}\widetilde{E}^{(1)}_{i\alpha}\widetilde{E}^{(2)*}_{i\alpha}\right|^{2}-\frac{1}{4}\sum_{i,\alpha}\left(E_{i\alpha}\right)^{4}$
(2.5)
$\displaystyle\to\left<I_{1}\right>\left<I_{2}\right>+\left|\int\text{d}k\int\text{d}\Omega\,\frac{\text{d}I}{\text{d}k\,\text{d}\Omega}e^{i\left[ck(t_{1}-t_{2})-k\hat{\boldsymbol{\mathbf{\theta}}}\cdot\boldsymbol{\mathbf{d}}\right]}\right|^{2}.$

For the first term in the second line, we have kept only “diagonal” terms from
the first line with $i_{1}=j_{1}$, $i_{2}=j_{2}$, $\alpha_{1}=\beta_{1}$, and
$\alpha_{2}=\beta_{2}$. For the second term, we kept “off-diagonal terms” with
$i_{1}=j_{2}$, $\alpha_{1}=\beta_{2}$, $i_{2}=j_{1}$, and
$\alpha_{2}=\beta_{1}$. The third term of the second line subtracts off terms
that we overcounted, but this is a negligible contribution in the continuum
limit taken in the third line of eq. 2.5. The second term in the third line
can be recognized as the modulus-squared of the complex visibility function.
We also introduced the baseline vector $\boldsymbol{\mathbf{d}}$, leading to a
difference in propagation distance $r_{i2}-r_{i1}$ to the two photodetectors:

$\displaystyle\boldsymbol{\mathbf{d}}\equiv\boldsymbol{\mathbf{r}}_{2}-\boldsymbol{\mathbf{r}}_{1}=-\boldsymbol{\mathbf{r}}_{i2}+\boldsymbol{\mathbf{r}}_{i1};\qquad
r_{i2}-r_{i1}=-\hat{\boldsymbol{\mathbf{\theta}}}_{i}\cdot\boldsymbol{\mathbf{d}},$
(2.6)

with the latter relation holding for all sources $i$.

We can write the “excess fractional intensity correlation” (relative to random
chance) as:

$\displaystyle C(\boldsymbol{\mathbf{d}},\tau)$
$\displaystyle\equiv\frac{\left<I_{1}(t){I_{2}(t+\tau)}\right>}{\left<I_{1}\right>\left<I_{2}\right>}-1$
(2.7)
$\displaystyle=\int\frac{\text{d}^{2}b_{1}}{A_{1}}\int\frac{\text{d}^{2}b_{2}}{A_{2}}\int\frac{\text{d}(\Delta
t)}{\sqrt{2\pi}\sigma_{t}}e^{\frac{-(\Delta
t)^{2}}{2\sigma_{t}^{2}}}\times\left|\int\text{d}k\int\text{d}\Omega\,\frac{\text{d}\widetilde{I}}{\text{d}k\,\text{d}\Omega}e^{i\left[k\left(c(\Delta
t-\tau)-\hat{\boldsymbol{\mathbf{\theta}}}\cdot(\boldsymbol{\mathbf{d}}+\boldsymbol{\mathbf{\Delta
b}})\right)\right]}\right|^{2},$

with $\text{d}\widetilde{I}\equiv\text{d}I/\left<I\right>$, shown for the
example of a single source in figure 3. In the second line of eq. 2.7, we have
introduced two “averaging kernels” over $\boldsymbol{\mathbf{b}}_{1}$ and
$\boldsymbol{\mathbf{b}}_{2}$ that take into account the finite aperture sizes
$A_{1}$ and $A_{2}$ of the telescopes associated with photodetectors 1 and 2,
effectively smearing out the baseline $\boldsymbol{\mathbf{d}}$ between the
centers of the telescopes by an amount $\boldsymbol{\mathbf{\Delta
b}}\equiv\boldsymbol{\mathbf{b}}_{2}-\boldsymbol{\mathbf{b}}_{1}$ for each
pair of points $\boldsymbol{\mathbf{b}}_{1}$ and $\boldsymbol{\mathbf{b}}_{2}$
on the respective telescopes’ apertures. A third averaging kernel over $\Delta
t$ takes into account the relative timing error of the clocks associated with
photodetectors $1$ and $2$, around their mean offset of $\tau\equiv
t_{2}-t_{1}$. Its timing spread $\sigma_{t}$ is

$\displaystyle\sigma_{t}^{2}=\sigma_{t_{\text{res},1}}^{2}+\sigma_{t_{\text{res},2}}^{2}+\sigma_{t_{\text{sync}}}^{2},$
(2.8)

the (quadrature) sum of the respective timing resolutions and the relative
synchronization error $\sigma_{t_{\text{sync}}}$.

For a sufficiently narrow frequency spectrum around $k=\overline{k}$, and with
negligible smearing and phase noise, the excess correlation of eq. 2.7 is
proportional to the _square modulus of the Fourier transform of the image_ ,
at an angular wavevector of $k\boldsymbol{\mathbf{d}}_{\perp}$, thus providing
a fiducial angular resolution

$\displaystyle\sigma_{\theta_{\text{res}}}\equiv\frac{1}{\overline{k}d}=\frac{\overline{\lambda}}{2\pi
d}\approx\underbrace{7.96\times
10^{-12}\,\text{rad}}_{1.64\,\mu\text{as}}\left(\frac{\overline{\lambda}}{500\,\text{nm}}\right)\left(\frac{10\,\text{km}}{d}\right).$
(2.9)

in the direction parallel to
$\boldsymbol{\mathbf{d}}_{\perp}\equiv\boldsymbol{\mathbf{d}}(1-\hat{\boldsymbol{\mathbf{d}}}\cdot\hat{\overline{\boldsymbol{\mathbf{\theta}}}})$
(with $\hat{\overline{\boldsymbol{\mathbf{\theta}}}}$ the location of the
light centroid of the image). This resolution is much better than that of
very-long-baseline amplitude interferometry in the radio band (e.g. EHT [28])
at much longer wavelengths (typically $\lambda\sim 1\,\text{mm}$), and than
that of optical/infrared Michelson interferometry (e.g. Keck [29], CHARA [26],
GRAVITY [69]) which is restricted to much shorter baselines ($d\lesssim
300\,\mathrm{m}$) because the light from both apertures needs to be physically
recombined. With intensity interferometry, the correlation can be computed
offline, and the detectors can in principle be placed at opposite sides of the
Earth. However, as we shall see in sections 2.3 and 2.4, for $d\gg
10\,\text{km}$ and naively even better angular resolution, the excess
correlation will be suppressed because all sufficiently bright sources for
intensity interferometry have an angular size larger than
$10^{-12}\,\text{rad}$, so the fiducial estimate of eq. 2.9 is (practically) a
best-case scenario.

####  2.1.2 Examples

Before we move on to extended-path intensity correlation (EPIC) in section
2.2, it is instructive to consider a handful of examples to gain intuition
about intensity interferometry and to understand its limitation without the
path extension.

The simplest example is that of a single point-like source $a$ at a location
$\hat{\boldsymbol{\mathbf{\theta}}}_{a}$ on the celestial sphere with a narrow
gaussian spectrum of width $\sigma_{k}$ around a mean wavenumber
$\overline{k}$:

$\displaystyle\frac{\text{d}I}{\text{d}k\,\text{d}\Omega}=\frac{I(\overline{k})}{\sqrt{2\pi}\sigma_{k}}e^{\frac{-(k-\overline{k})^{2}}{2\sigma_{k}^{2}}}\delta^{2}\left(\hat{\boldsymbol{\mathbf{\theta}}}-\hat{\boldsymbol{\mathbf{\theta}}}_{a}\right).$
(2.10)

Assuming that the planes of the apertures are oriented (sufficiently)
perpendicular to the location of the source, i.e.
$|\boldsymbol{\mathbf{b}}_{p}\cdot\hat{\boldsymbol{\mathbf{\theta}}}_{a}|\ll
c\sigma_{t}$, the aperture smearing can be ignored, and we find after
evaluating eq. 2.7:

$\displaystyle C(\boldsymbol{\mathbf{d}},\tau)$
$\displaystyle=\int\text{d}(\Delta
t)\frac{1}{\sqrt{2\pi}\sigma_{t}}e^{\frac{-(\Delta
t)^{2}}{2\sigma_{t}^{2}}}\left|\exp\left\\{i\overline{k}\left(c(\Delta
t-\tau)-\hat{\boldsymbol{\mathbf{\theta}}}_{a}\cdot\boldsymbol{\mathbf{d}}\right)-\frac{\sigma_{k}^{2}}{2}\left(c(\Delta
t-\tau)-\hat{\boldsymbol{\mathbf{\theta}}}_{a}\cdot\boldsymbol{\mathbf{d}}\right)^{2}\right\\}\right|^{2}$
$\displaystyle=\frac{1}{\sqrt{1+2c^{2}\sigma_{k}^{2}\sigma_{t}^{2}}}\exp\left\\{-\frac{\sigma_{k}^{2}}{1+2c^{2}\sigma_{k}^{2}\sigma_{t}^{2}}\left(c\tau+\hat{\boldsymbol{\mathbf{\theta}}}_{a}\cdot\boldsymbol{\mathbf{d}}\right)^{2}\right\\}.$
(2.11)

Thus, we find that the intensity fluctuations at the two photodetectors are
maximally correlated as long as the timing offset is chosen to be
$\tau\simeq-\hat{\boldsymbol{\mathbf{\theta}}}_{a}\cdot\boldsymbol{\mathbf{d}}/c$,
commensurate to the differential wavefront arrival time at both
photodetectors. Practically, the coherence time of the light is significantly
shorter than the relative timing precision, so $c\sigma_{k}\sigma_{t}\gg 1$,
as we will discuss in section 2.4. Hence, the offset $\tau$ needs to be chosen
around its optimal value to a precision of $\sigma_{t}$ or better, which then
produces a maximal intensity correlation of
$C(\boldsymbol{\mathbf{d}},\tau=-\hat{\boldsymbol{\mathbf{\theta}}}_{a}\cdot\boldsymbol{\mathbf{d}}/c)\simeq
1/(\sqrt{2}c\sigma_{k}\sigma_{t})$.

For our next example, let us take two point-like sources $a$ and $b$ at
locations $\hat{\boldsymbol{\mathbf{\theta}}}_{a}$ and
$\hat{\boldsymbol{\mathbf{\theta}}}_{b}$ with total flux
$\left<I\right>=I(\overline{k})$ and fractional fluxes $\widetilde{I}_{a}$ and
$\widetilde{I}_{b}$ (by construction,
$\widetilde{I}_{a}+\widetilde{I}_{b}=1$):

$\displaystyle\frac{\text{d}\widetilde{I}}{\text{d}k\,\text{d}\Omega}=\frac{1}{\sqrt{2\pi}\sigma_{k}}e^{\frac{-(k-\overline{k})^{2}}{2\sigma_{k}^{2}}}\left[\sum_{i=a,b}\widetilde{I}_{i}\,\delta^{2}\left(\hat{\boldsymbol{\mathbf{\theta}}}-\hat{\boldsymbol{\mathbf{\theta}}}_{i}\right)\right].$
(2.12)

We will again neglect the finite aperture spread, despite the fact that
$\hat{\boldsymbol{\mathbf{\theta}}}_{i}\cdot\boldsymbol{\mathbf{b}}_{p}=0$
cannot be maintained for both sources simultaneously, as more severe
suppression effects will typically kick in for
$\boldsymbol{\mathbf{\theta}}_{ba}\equiv\hat{\boldsymbol{\mathbf{\theta}}}_{b}-\hat{\boldsymbol{\mathbf{\theta}}}_{a}\neq
0$. We evaluate eq. 2.7:

$\displaystyle C(\boldsymbol{\mathbf{d}},\tau)$
$\displaystyle=\int\text{d}(\Delta
t)\frac{1}{\sqrt{2\pi}\sigma_{t}}e^{\frac{-(\Delta
t)^{2}}{2\sigma_{t}^{2}}}\left|\sum_{i=a,b}\widetilde{I}_{i}\,e^{i\overline{k}\left(c(\Delta
t-\tau)-\hat{\boldsymbol{\mathbf{\theta}}}_{i}\cdot\boldsymbol{\mathbf{d}}\right)-\frac{\sigma_{k}^{2}}{2}\left(c(\Delta
t-\tau)-\hat{\boldsymbol{\mathbf{\theta}}}_{i}\cdot\boldsymbol{\mathbf{d}}\right)^{2}}\right|^{2}$
$\displaystyle=\frac{1}{\sqrt{1+2c^{2}\sigma_{k}^{2}\sigma_{t}^{2}}}\Bigg{\\{}\widetilde{I}_{a}^{2}e^{\frac{-\sigma_{k}^{2}}{1+2c^{2}\sigma_{k}^{2}\sigma_{t}^{2}}\left(c\tau+\hat{\boldsymbol{\mathbf{\theta}}}_{a}\cdot\boldsymbol{\mathbf{d}}\right)^{2}}+\widetilde{I}_{b}^{2}e^{\frac{-\sigma_{k}^{2}}{1+2c^{2}\sigma_{k}^{2}\sigma_{t}^{2}}\left(c\tau+\hat{\boldsymbol{\mathbf{\theta}}}_{b}\cdot\boldsymbol{\mathbf{d}}\right)^{2}}$
(2.13)
$\displaystyle\phantom{=}+2\widetilde{I}_{a}\widetilde{I}_{b}\cos\left[\overline{k}\boldsymbol{\mathbf{d}}\cdot\boldsymbol{\mathbf{\theta}}_{ba}\right]e^{-\sigma_{k}^{2}(\boldsymbol{\mathbf{\theta}}_{ba}\cdot\boldsymbol{\mathbf{d}})^{2}/4}e^{\frac{-\sigma_{k}^{2}}{1+2c^{2}\sigma_{k}^{2}\sigma_{t}^{2}}\left[c\tau+(\hat{\boldsymbol{\mathbf{\theta}}}_{a}+\hat{\boldsymbol{\mathbf{\theta}}}_{b})\cdot\boldsymbol{\mathbf{d}}/2\right]^{2}}\Bigg{\\}}$

The generalization of eq. 2.13 to an arbitrary number of point-like sources is
straightforward and has a similar form.

We find that the excess correlation of eq. 2.13 has “one-source”
contributions, proportional to $\widetilde{I}_{i}^{2}$, of exactly the same
functional form as in eq. 2.11. The last line of eq. 2.13 contains the “two-
source” contribution, proportional to $\widetilde{I}_{a}\widetilde{I}_{b}$.
This is the term most sensitive to the separation of the sources, as it
contains the most rapidly varying phase, namely
$\overline{k}\boldsymbol{\mathbf{d}}\cdot\boldsymbol{\mathbf{\theta}}_{ba}$ in
the cosine argument. The two-source contribution is maximized when the timing
offset is taken to be
$\tau=\tau^{\text{opt}}=-(\hat{\boldsymbol{\mathbf{\theta}}}_{a}+\hat{\boldsymbol{\mathbf{\theta}}}_{b})\cdot\boldsymbol{\mathbf{d}}/2c$,
and we find in the limit $c\sigma_{k}\sigma_{t}\gg 1$:

$\displaystyle
C^{\text{opt}}(\boldsymbol{\mathbf{d}})=\frac{1}{\sqrt{2}c\sigma_{k}\sigma_{t}}\Bigg{\\{}(\widetilde{I}_{a}^{2}+\widetilde{I}_{b}^{2})e^{\frac{-(\boldsymbol{\mathbf{\theta}}_{ba}\cdot\boldsymbol{\mathbf{d}})^{2}}{8\sigma_{t}^{2}}}+2\widetilde{I}_{a}\widetilde{I}_{b}\cos\left[\overline{k}\boldsymbol{\mathbf{d}}\cdot\boldsymbol{\mathbf{\theta}}_{ba}\right]e^{\frac{-\sigma_{k}^{2}(\boldsymbol{\mathbf{\theta}}_{ba}\cdot\boldsymbol{\mathbf{d}})^{2}}{4}}\Bigg{\\}}.$
(2.14)

In the above eq. 2.14, we recognize three angular scales. The first and
smallest is the angular resolution scale
$\sigma_{\theta_{\text{res}}}=1/\overline{k}d$ from eq. 2.9, which sets the
fringe separation and thus determines the angular resolution on
$\boldsymbol{\mathbf{\theta}}_{ba}\cdot\hat{\boldsymbol{\mathbf{d}}}$. The
second angular scale is

$\displaystyle\sigma_{\Delta\theta}=\frac{\sqrt{2}}{\sigma_{k}d}\approx\underbrace{5.63\times
10^{-8}\,\text{rad}}_{11.6\,\text{mas}}\left(\frac{\overline{k}/\sigma_{k}}{1{,}000}\right)\left(\frac{\overline{\lambda}}{500\,\text{nm}}\right)\left(\frac{10\,\text{km}}{d}\right),$
(2.15)

which sets the dynamical range for
$\boldsymbol{\mathbf{\theta}}_{ba}\cdot\hat{\boldsymbol{\mathbf{d}}}$ over
which the fringes are not exponentially suppressed, i.e. the effective field
of view. The third and typically largest angular scale is:

$\displaystyle\sigma_{\hat{\boldsymbol{\mathbf{\theta}}}}=\frac{2c\sigma_{t}}{d}\approx\underbrace{6.00\times
10^{-7}\,\text{rad}}_{124\,\text{mas}}\left(\frac{\sigma_{t}}{10\,\mathrm{ps}}\right)\left(\frac{10\,\text{km}}{d}\right),$
(2.16)

which determines the fiducial, _global_ astrometric precision on
$\hat{\boldsymbol{\mathbf{\theta}}}_{i}\cdot\hat{\boldsymbol{\mathbf{d}}}$
from the “one-source” contributions to the excess intensity correlation,
independent of any pairwise angular separations
$\boldsymbol{\mathbf{\theta}}_{ij}$. In figure 4, we plot the excess
correlation of eq. 2.14 for two equally bright sources, and indicate the three
angular scales from eqs. 2.9, 2.15, and 2.16 for some representative
parameters.

(a)

(b)

Figure 4: Excess correlation $C^{\text{opt}}(\boldsymbol{\mathbf{d}})$ as a
function of
$\boldsymbol{\mathbf{\theta}}_{ba}\cdot\hat{\boldsymbol{\mathbf{d}}}$ for
fiducial parameters of the baseline separation $d=10\,\text{km}$, mean
wavenumber $\overline{k}=2\pi/500\,\text{nm}$ of the light, and relative
timing precision $\sigma_{t}=30\,\text{ps}$, for two equally bright sources:
$\widetilde{I}_{a}=\widetilde{I}_{b}=1/2$. The blue curve in figure 4(a) is
eq. 2.14 for intensity interferometry _without_ any path extension. The red
curve (spike) in figure 4(a) is the excess correlation of eq. 2.28 for EPIC,
with the same parameters but including geometric delays of
$\ell_{1}=\ell_{2}=10\,\text{cm}$, producing a ghost intensity correlation
fringe of $1/4$ the contrast at a reference angle
$\boldsymbol{\mathbf{\theta}}_{ba}^{\text{ref}}\cdot\hat{\boldsymbol{\mathbf{d}}}=2\times
10^{-5}\,\text{rad}$, but otherwise identical to the main fringe. A zoom-in of
this ghost fringe around
$\boldsymbol{\mathbf{\theta}}_{ba}\approx\boldsymbol{\mathbf{\theta}}_{ba}^{\text{ref}}$
is shown in figure 4(b). The characteristic scales for angular resolution
$\sigma_{\theta_{\text{res}}}$ (eq. 2.9), dynamic field of view
$\sigma_{\Delta\theta}$ (eq. 2.15), and global astrometric precision
$\sigma_{\hat{\boldsymbol{\mathbf{\theta}}}}$ (eq. 2.16) are also indicated.

###  2.2 Extended-path intensity correlation

####  2.2.1 General

The limited (angular) dynamic range of intensity interferometry severely
limits its use for differential astrometric measurements. If one wants to
harness the full resolving power at a resolution
$\sigma_{\theta_{\text{res}}}\sim\mathcal{O}(1\,\mu\text{as})$ (see eq. 2.9),
the effective field of view is also tiny, of order
$\sigma_{\Delta\theta}\lesssim\mathcal{O}(10\,\text{mas})$ if one has no more
than $\overline{k}/\sigma_{k}\sim 10^{4}$ fringes (see eq. 2.15), which is
already restrictive on the spectral width. Very few bright sources, other than
tight binaries, are expected have such small relative separations.
Furthermore, it is practically difficult to discriminate between the
$n^{\text{th}}$ fringe and the $(n\pm 1)^{\text{th}}$ fringe for large $n\sim
k\boldsymbol{\mathbf{d}}\cdot\boldsymbol{\mathbf{\theta}}_{ba}/2\pi\gg 1$.

(a) Light gathering and collimation

(b) Path extension and spectral splitting

Figure 5: Schematic optical path for the path extension in a single telescope
of an extended-path intensity correlator. Panel 5(a): Light from a compact
source is collected by a large primary mirror with focal length $f$, and
collimated down to a narrow beam by a secondary mirror with focal length
$f^{\prime}$ sharing the same focal point $F$ as the primary. (Actual setups
may have several foci, and a slit near a real pupil to select light only from
the source of interest.) Panel 5(b): A semi-transparent flat plate whose
normal is at an angle $\gamma$ relative to the collimated beam, splits the
light into two optical paths (gray and black) of different path lengths (for
$\gamma\simeq\gamma^{\prime}\neq 45^{\circ}$) but undistorted wavefronts (for
$\gamma=\gamma^{\prime}$). The paths are recombined by another semi-
transparent plate, after which the collimated beam is spectrally split by a
reflective diffraction grating, and focused onto a photodetector whose pixels
are read out as different spectral channels.

EPIC solves this small-separation problem, and allows for effective fields of
view on the order of a _few arcseconds_. It is accomplished by modifying the
optical path in at least one of the telescopes by introducing a differential
path extension illustrated in figure 5. Light from the source(s) impinges on
the telescope’s large primary mirror (with collecting area $A_{p}$), and is
then re-collimated to a narrow beam (figure 5(a)). This collimated light then
enters the path-extension stage (top half of figure 5(b)), where the beam is
split and recombined by a pair of partially reflective plates, with electric-
field reflectivity and transmission of $\epsilon$ and $\sqrt{1-\epsilon^{2}}$,
respectively. For $\gamma=\gamma^{\prime}$, the two paths have a differential
geometric path length of:

$\displaystyle\ell_{p}=\frac{w_{p}}{\sin 2\gamma}\left(1+\cos 2\gamma-\sin
2\gamma\right)\quad\text{for }0<\gamma<\pi/2,$ (2.17)

which can be positive ($\gamma<\pi/4$) or negative ($\gamma>\pi/4$). It will
be advantageous to have $\gamma^{\prime}$ deviate slightly from $\gamma$, as
discussed around eqs. 2.25 and 2.26. After the path-extension stage, the light
is spectrally split (bottom half of right panel), and focused onto a
photodetector array, whose individual pixels each read out one spectral
channel, centered at some wavenumber $\overline{k}$ with narrow bandwidth
$\sigma_{k}$. We discuss photodetector capabilities in section 3.3 and
spectroscopic elements in section 3.4.

In such a setup, with optical losses and plate thicknesses ignored, the
electric field as seen at the location of (a single pixel of) photodetector
$p$ is:

$\displaystyle E_{p}(t_{p})$
$\displaystyle=\epsilon\sqrt{1-\epsilon^{2}}\sum_{i=1}^{N_{\theta}}\sum_{\alpha=1}^{N_{k}}\sum_{n=0}^{1}e^{-ink_{\alpha}\ell_{p}}E_{i\alpha}\exp\left\\{i\left[k_{\alpha}(t_{p}-r_{ip})+\phi^{\text{em}}_{i\alpha}\right]\right\\},$
(2.18)

where $n=0$ corresponds to the “left path” in figure 5(b), and $n=1$ to the
“right path”, which have a differential geometric propagation phase due to
their difference in length by $\ell_{p}$ from eq. 2.17. The expected intensity
calculation proceeds exactly as in eq. 2.4, and becomes in the continuum
limit:

$\displaystyle\left<I_{p}(t_{p})\right>=2\epsilon^{2}\left(1-\epsilon^{2}\right)\int\text{d}k\int\text{d}\Omega\,\frac{\text{d}I}{\text{d}k\,\text{d}\Omega}\left(1+\cos
k\ell_{p}\right).$ (2.19)

The latter factor $(1+\cos k\ell_{p})$ is due to interference of the left and
right paths: it is maximal when the $\ell_{p}$ is an integer number of
wavelengths, and vanishes when $\ell_{p}$ is a half-integer number of
wavelengths (corresponding to when the photodetector is at the “dark port” of
the internal interferometer). The calculation of the intensity correlation is
analogous to that of eq. 2.5, with the addition of a
$\epsilon\sqrt{1-\epsilon^{2}}(1+e^{-ik\ell_{p}})$ factor for every electric
field $E_{p}$:

$\displaystyle\left<I_{1}(t_{1})I_{2}(t_{2})\right>=\left<I_{1}\right>\left<I_{2}\right>+\epsilon^{4}\left(1-\epsilon^{2}\right)^{2}$
(2.20)
$\displaystyle\times\left|\int\text{d}k\int\text{d}\Omega\,\frac{\text{d}I}{\text{d}k\,\text{d}\Omega}\big{(}1+e^{-ik\ell_{1}}\big{)}\big{(}1+e^{ik\ell_{2}}\big{)}e^{ik\left((t_{1}-t_{2})-\hat{\boldsymbol{\mathbf{\theta}}}\cdot\boldsymbol{\mathbf{d}}\right)}\right|^{2}.$

Putting together eqs. 2.7, 2.19, and 2.20, and including aperture smearing and
finite relative timing resolution effects, thus yields the excess fractional
intensity correlation:

$\displaystyle
C(\boldsymbol{\mathbf{d}},\tau)=\frac{\epsilon^{4}\left(1-\epsilon^{2}\right)^{2}}{\left<I_{1}\right>\left<I_{2}\right>}\int\frac{\text{d}^{2}b_{1}}{A_{1}}\int\frac{\text{d}^{2}b_{2}}{A_{2}}\int\text{d}(\Delta
t)\frac{1}{\sqrt{2\pi}\sigma_{t}}e^{\frac{-(\Delta t)^{2}}{2\sigma_{t}^{2}}}$
(2.21)
$\displaystyle\times\left|\int\text{d}k\int\text{d}\Omega\,\frac{\text{d}I}{\text{d}k\,\text{d}\Omega}\big{(}1+e^{-ik\ell_{1}}\big{)}\big{(}1+e^{ik\ell_{2}}\big{)}e^{ik\left(c(\Delta
t-\tau)-\hat{\boldsymbol{\mathbf{\theta}}}\cdot(\boldsymbol{\mathbf{d}}+\boldsymbol{\mathbf{\Delta
b}})\right)}\right|^{2}.$

Equation 2.21 is the central formula for extended-path intensity correlation.

####  2.2.2 Examples

To understand the consequences of the path extension in eq. 2.21, let us again
consider a single point source with a differential intensity distribution from
eq. 2.10. The effective transmitted intensity at each telescope is, from eq.
2.19:

$\displaystyle\left<I_{p}\right>=2\epsilon^{2}(1-\epsilon^{2})I(\overline{k})\Big{[}1+e^{-\sigma_{k}^{2}\ell_{p}^{2}/2}\cos\overline{k}\ell_{p}\Big{]}\simeq
2\epsilon^{2}(1-\epsilon^{2})I(\overline{k}).$ (2.22)

As we will see shortly, to effectively increase the field of view EPIC beyond
that of conventional intensity interferometry, it is necessary to take
$\sigma_{k}\ell\gg 1$, so that the last approximation in eq. 2.22 will hold to
high precision. The excess intensity correlation for this source can be
calculated from eq. 2.20, with the same approximation and with apertures
perpendicular to the line of sight
($\hat{\boldsymbol{\mathbf{\theta}}}_{a}\cdot\boldsymbol{\mathbf{b}}_{p}$):

$\displaystyle C(\boldsymbol{\mathbf{d}},\tau)$
$\displaystyle\simeq\frac{e^{\frac{-\sigma_{k}^{2}\left(c\tau+\hat{\boldsymbol{\mathbf{\theta}}}_{a}\cdot\boldsymbol{\mathbf{d}}\right)^{2}}{1+2c^{2}\sigma_{k}^{2}\sigma_{t}^{2}}}+e^{\frac{-\sigma_{k}^{2}\left(c\tau+\hat{\boldsymbol{\mathbf{\theta}}}_{a}\cdot\boldsymbol{\mathbf{d}}+\ell_{1}\right)^{2}}{1+2c^{2}\sigma_{k}^{2}\sigma_{t}^{2}}}+e^{\frac{-\sigma_{k}^{2}\left(c\tau+\hat{\boldsymbol{\mathbf{\theta}}}_{a}\cdot\boldsymbol{\mathbf{d}}-\ell_{2}\right)^{2}}{1+2c^{2}\sigma_{k}^{2}\sigma_{t}^{2}}}+e^{\frac{-\sigma_{k}^{2}\left(c\tau+\hat{\boldsymbol{\mathbf{\theta}}}_{a}\cdot\boldsymbol{\mathbf{d}}-\ell_{2}+\ell_{1}\right)^{2}}{1+2c^{2}\sigma_{k}^{2}\sigma_{t}^{2}}}}{4\sqrt{1+2c^{2}\sigma_{k}^{2}\sigma_{t}^{2}}}.$
(2.23)

Comparing eqs. 2.11 and 2.23, or the black and orange curves in figure 2, we
can see that the effect of the path extension is to reduce the contrast of the
“main” intensity fringe, at
$c\tau\simeq-\hat{\boldsymbol{\mathbf{\theta}}}\cdot\boldsymbol{\mathbf{d}}$,
by a factor of 4. More importantly, however, the initial fringe is now also
split into four, including three other equally bright “ghost fringes”, as if
there were three duplicate images of the source at locations
$\hat{\boldsymbol{\mathbf{\theta}}}_{a}^{\text{ghost}}=\hat{\boldsymbol{\mathbf{\theta}}}_{a}+\frac{\\{\ell_{1},-\ell_{2},-\ell_{2}+\ell_{1}\\}}{d}\hat{\boldsymbol{\mathbf{d}}}$,
respectively. When $\sigma_{k}\ell_{p}\gg 1$, these ghost fringes are far
outside the effective field of view (relative to the main fringe) of regular
intensity interferometry (cfr. eq. 2.15).

With the behavior of the extended-path intensity correlator understood for a
single point source, let us see what the excess intensity correlator looks
like for two point sources under influence of the path extension; evaluation
of the fractional intensity correlation formula of eq. 2.20 for the two
sources of eq. 2.12 gives:

$\displaystyle
C(\boldsymbol{\mathbf{d}},\tau)=\frac{1}{4\sqrt{1+2c^{2}\sigma_{k}^{2}\sigma_{t}^{2}}}\int\frac{\text{d}^{2}b_{1}}{A_{1}}\int\frac{\text{d}^{2}b_{2}}{A_{2}}\Bigg{\\{}\sum_{i=a,b}\widetilde{I}_{i}^{2}\sum_{\ell}e^{\frac{-\sigma_{k}^{2}}{1+2c^{2}\sigma_{k}^{2}\sigma_{t}^{2}}\left[c\tau+\hat{\boldsymbol{\mathbf{\theta}}}_{i}\cdot(\boldsymbol{\mathbf{d}}+\Delta\boldsymbol{\mathbf{b}})+\ell\right]^{2}}$
(2.24)
$\displaystyle+2\widetilde{I}_{a}\widetilde{I}_{b}\sum_{\ell,\ell^{\prime}}\cos\left[\overline{k}\left((\boldsymbol{\mathbf{d}}+\Delta\boldsymbol{\mathbf{b}})\cdot\boldsymbol{\mathbf{\theta}}_{ba}-\ell+\ell^{\prime}\right)\right]e^{\frac{-\sigma_{k}^{2}}{4}\left[\boldsymbol{\mathbf{\theta}}_{ba}\cdot(\boldsymbol{\mathbf{d}}+\Delta\boldsymbol{\mathbf{b}})-\ell+\ell^{\prime}\right]^{2}}e^{\frac{-\sigma_{k}^{2}\left[c\tau+\frac{\hat{\boldsymbol{\mathbf{\theta}}}_{a}+\hat{\boldsymbol{\mathbf{\theta}}}_{b}}{2}\cdot(\boldsymbol{\mathbf{d}}+\boldsymbol{\mathbf{\Delta
b}})+\frac{\ell+\ell^{\prime}}{2}\right]^{2}}{1+2c^{2}\sigma_{k}^{2}\sigma_{t}^{2}}}\Bigg{\\}}.$

Each sum over $\ell$ and $\ell^{\prime}$ runs over the list
$\\{0,\ell_{1},-\ell_{2},-\ell_{2}+\ell_{1}\\}$ with $\ell_{1}$ and $\ell_{2}$
as in eq. 2.17. The possibilities for $\ell$ correspond to whether the light
from source $a$ is extended nowhere, in telescope 1 only, in telescope 2 only,
or in both telescopes, respectively; $\ell^{\prime}$ runs over the same
possibilities for source $b$. For
$\boldsymbol{\mathbf{\theta}}_{ba}\cdot\hat{\boldsymbol{\mathbf{d}}}\gg\sigma_{\Delta\theta}\equiv\sqrt{2}/{\sigma_{k}d}$,
the “main” differential astrometry fringe between sources $a$ and $b$
(corresponding to $\ell=\ell^{\prime}=0$), or really any fringe with
$\sigma_{k}\left|\boldsymbol{\mathbf{\theta}}_{ba}\cdot\boldsymbol{\mathbf{d}}-\ell+\ell^{\prime}\right|\gg
1$, is exponentially suppressed (see eq. 2.15 and figure 2).

However, even when the separation between the main images of the source is
large, the apparent separation between their respective ghost images need not
be. By tuning
$\ell-\ell^{\prime}\simeq\boldsymbol{\mathbf{\theta}}_{ba}\cdot\hat{\boldsymbol{\mathbf{d}}}$,
we can overcome the field-of-view limitation of intensity interferometry.
Furthermore, by making $\ell_{p}$ depend on $\boldsymbol{\mathbf{b}}_{p}$, for
example by a slight tilting of one of the mirrors
($\gamma\neq\gamma^{\prime}$), aperture smearing effects can be negated
completely. The optimal geometric path extensions for differential astrometry
with EPIC are achieved at e.g. $\ell=+\ell_{1}^{\text{opt}}$ and
$\ell^{\prime}=-\ell_{2}^{\text{opt}}$ with:

$\displaystyle\ell_{1}^{\text{opt}}(\boldsymbol{\mathbf{b}}_{1})$
$\displaystyle=\boldsymbol{\mathbf{\theta}}^{\text{ref}}_{ba}\cdot\big{[}\rho\boldsymbol{\mathbf{d}}-\boldsymbol{\mathbf{b}}_{1}\big{]},$
(2.25) $\displaystyle\ell_{2}^{\text{opt}}(\boldsymbol{\mathbf{b}}_{2})$
$\displaystyle=\boldsymbol{\mathbf{\theta}}_{ba}^{\text{ref}}\cdot\big{[}(1-\rho)\boldsymbol{\mathbf{d}}+\boldsymbol{\mathbf{b}}_{2}\big{]};$
(2.26)

where $\rho$ is arbitrary, and other fringe combinations of $\ell$ and
$\ell^{\prime}$ are also possible. This particular fringe corresponds to the
amplitude depicted in figure 2. The reference separation vector
$\boldsymbol{\mathbf{\theta}}_{ba}^{\text{ref}}$ should be chosen close to the
actual separation, i.e.
$\delta\boldsymbol{\mathbf{\theta}}_{ba}\equiv\boldsymbol{\mathbf{\theta}}_{ba}-\boldsymbol{\mathbf{\theta}}_{ba}^{\text{ref}}$
has length much less than $\sigma_{\Delta\theta}$. For maximum sensitivity,
one should choose the optimal timing offset $\tau=\tau^{\text{opt}}$ that
maximizes the differential astrometry fringe:

$\displaystyle\tau^{\text{opt}}=-\frac{\hat{\boldsymbol{\mathbf{\theta}}}_{a}+\hat{\boldsymbol{\mathbf{\theta}}}_{b}-(1-2\rho)\boldsymbol{\mathbf{\theta}}^{\text{ref}}_{ba}}{2}\cdot\frac{\boldsymbol{\mathbf{d}}}{c}.$
(2.27)

Finally, we obtain the excess fractional intensity correlation for two point
sources in this optimal configuration:

$\displaystyle
C^{\text{opt}}(\boldsymbol{\mathbf{d}})\simeq\frac{1}{4\sqrt{2}c\sigma_{k}\sigma_{t}}$
$\displaystyle\Bigg{\\{}\widetilde{I}_{a}^{2}+\widetilde{I}_{b}^{2}+2\widetilde{I}_{a}\widetilde{I}_{b}\cos\left[\overline{k}\boldsymbol{\mathbf{d}}\cdot\boldsymbol{\mathbf{\delta\theta}}_{ba}\right]e^{\frac{-\sigma_{k}^{2}}{4}\left(\boldsymbol{\mathbf{d}}\cdot\boldsymbol{\mathbf{\delta\theta}}_{ba}\right)^{2}}\Bigg{\\}}.$
(2.28)

The utility of EPIC is apparent upon comparison of eqs. 2.14 and 2.28. Without
the path extension, the two-source fringes are only visible for an ultra-
narrow field:
$|\boldsymbol{\mathbf{\theta}}_{ba}\cdot\hat{\boldsymbol{\mathbf{d}}}|\lesssim\sigma_{\Delta\theta}$.
With the path extension, the two-source fringes are visible whenever
$|\boldsymbol{\mathbf{\delta\theta}}_{ba}\cdot\hat{\boldsymbol{\mathbf{d}}}|\lesssim\sigma_{\Delta\theta}$.
In other words, the dynamic range has not increased, but instead of the usable
field of view being centered around the locus where
$\boldsymbol{\mathbf{\theta}}_{ba}\cdot\hat{\boldsymbol{\mathbf{d}}}\approx
0$, it is centered around
$\boldsymbol{\mathbf{\theta}}_{ba}\cdot\hat{\boldsymbol{\mathbf{d}}}\approx\boldsymbol{\mathbf{\theta}}^{\text{ref}}_{ba}\cdot\hat{\boldsymbol{\mathbf{d}}}=\frac{\ell_{1}+\ell_{2}}{d}$.
From the point of view of the excess intensity correlation, the path
extensions resulting from the beamsplitter in each telescopes according to
eqs. 2.25 and 2.26 effectively produce _co-located ghost images_ of sources
$a$ and $b$, despite their true images located at different points on the
celestial sphere. Furthermore, the $\boldsymbol{\mathbf{b}}_{p}$-dependence of
$\ell^{\text{opt}}_{b}$ can correct for the aperture smearing at each
telescope.

###  2.3 Finite source size

In sections 2.1.2 and 2.2.2, we have given examples of intensity correlation
signals for _point-like_ sources. However, once the intensity interferometer
resolves either or both of the sources
($\sigma_{\theta_{\text{res}}}\lesssim\theta_{j}$ with $\theta_{j}$ the
characteristic angular size of the source), the visibility is reduced. This is
an important effect, because the differential light-centroiding precision
between two sources scales favorably with baseline distance $d$ until the
sources are resolved, so the optimal operating point will be
$\sigma_{\theta_{\text{res}}}\sim\theta_{j}$ (see section 2.4). We review the
details of this finite-source-size effect on the intensity fringes here for
the two main sources of interest—stars and quasars—but refer the reader to the
classic references [35, 36] for more background, as determining stellar radii
has been the primary use case of intensity interferometry so far.

We write the differential surface brightness per unit wavenumber $k$ in a
single spectral channel of bandwidth $\sigma_{k}$ centered at $k=\overline{k}$
as:

$\displaystyle\frac{\text{d}I}{\text{d}k\,\text{d}\Omega}=e^{\frac{-(k-\overline{k})^{2}}{2\sigma_{k}^{2}}}\sum_{j}^{\text{sources}}\left.\frac{\text{d}I_{j}}{\text{d}k\,\text{d}\Omega}\right|_{\text{thermal}},$
(2.29)

where the pre-factor approximates the effect of a (Gaussian) spectral filter,
and we assume that the sources $j$ are optically thick thermal emitters, whose
intrinsic differential surface brightness

$\displaystyle\left.\frac{\text{d}I_{j}}{\text{d}k\,\text{d}\Omega}\right|_{\text{thermal}}=\frac{\hbar
c^{2}}{4\pi^{3}}\frac{k^{3}}{e^{\hbar
ck/k_{\mathrm{B}}T_{j}(\hat{\boldsymbol{\mathbf{\theta}}}-\hat{\boldsymbol{\mathbf{\theta}}}_{j})}-1}$
(2.30)

depends only on the temperature function
$T_{j}(\hat{\boldsymbol{\mathbf{\theta}}}-\hat{\boldsymbol{\mathbf{\theta}}}_{j})$
with $\hat{\boldsymbol{\mathbf{\theta}}}_{j}$ taken to be the light centroid
of the source and $k_{\mathrm{B}}$ the Boltzmann constant. The excess
fractional intensity correlation signals of eqs. 2.7 and 2.21 contain
integrals of the form:

$\displaystyle\int\text{d}k\int\text{d}\Omega\,\frac{\text{d}I}{\text{d}k\,\text{d}\Omega}\,e^{-ik\hat{\boldsymbol{\mathbf{\theta}}}\cdot\boldsymbol{\mathbf{d}}}\simeq
I(\overline{k})\int\text{d}k\,\frac{e^{\frac{-(k-\overline{k})^{2}}{2\sigma_{k}^{2}}}}{\sqrt{2\pi}\sigma_{k}}\,\sum_{j}\,e^{-ik\hat{\boldsymbol{\mathbf{\theta}}}_{j}\cdot\boldsymbol{\mathbf{d}}}\,\widetilde{I}_{j}(\overline{k})\,\mathcal{F}_{j}(\overline{k},\boldsymbol{\mathbf{d}})$
(2.31)

where we have defined the total flux $I(\overline{k})$ in the channel centered
at $\overline{k}$

$\displaystyle
I(\overline{k})\equiv\sum_{j}I_{j}(\overline{k})=\sum_{j}\int\text{d}\Omega\,\frac{\hbar
c^{2}}{2^{3/2}\pi^{5/2}}\frac{\sigma_{k}\overline{k}^{3}}{e^{\hbar
c\overline{k}/k_{\mathrm{B}}T_{j}(\hat{\boldsymbol{\mathbf{\theta}}})}-1},$
(2.32)

and the fractional fluxes $\widetilde{I}_{j}(\overline{k})\equiv
I_{j}(\overline{k})/I(\overline{k})$. The “form factor” is

$\displaystyle\mathcal{F}_{j}(\overline{k},\boldsymbol{\mathbf{d}})\equiv\frac{\int\text{d}\Omega\,e^{-i\overline{k}\boldsymbol{\mathbf{d}}\cdot\hat{\boldsymbol{\mathbf{\theta}}}}\left[e^{\hbar
c\overline{k}/k_{\mathrm{B}}T_{j}(\hat{\boldsymbol{\mathbf{\theta}}})}-1\right]^{-1}}{\int\text{d}\Omega\left[e^{\hbar
c\overline{k}/k_{\mathrm{B}}T_{j}(\hat{\boldsymbol{\mathbf{\theta}}})}-1\right]^{-1}}$
(2.33)

and is equal to unity for point-like sources, i.e.
$\mathcal{F}_{j}(\overline{k},\boldsymbol{\mathbf{d}})\to 1$ as
$\overline{k}d\theta_{j}\to 1$, and is the implicit limit we have worked in
for the examples in sections 2.1.2 and 2.2.2.

Incorporating finite-size effects is straightforward upon inspection of eq.
2.31: for every (fractional) flux factor $\widetilde{I}_{j}(\overline{k})$ of
a source $j$ in a spectral channel centered at $\overline{k}$, one is to
insert also the form factor
$\mathcal{F}_{j}(\overline{k},\boldsymbol{\mathbf{d}})$ from eq. 2.33. For
example, the excess fractional correlation for two sources $a$ and $b$ from
eq. 2.28 is modified to:

$\displaystyle C^{\text{opt}}(\boldsymbol{\mathbf{d}})$
$\displaystyle\simeq\frac{1}{4\sqrt{2}c\sigma_{k}\sigma_{t}}\Bigg{\\{}\widetilde{I}_{a}(\overline{k})^{2}\left|\mathcal{F}_{a}(\overline{k},\boldsymbol{\mathbf{d}})\right|^{2}+\widetilde{I}_{b}(\overline{k})^{2}\left|\mathcal{F}_{b}(\overline{k},\boldsymbol{\mathbf{d}})\right|^{2}$
(2.34)
$\displaystyle\phantom{\simeq}+2\,\widetilde{I}_{a}(\overline{k})\,\widetilde{I}_{b}(\overline{k})\,\text{Re}\left\\{\mathcal{F}_{a}(\overline{k},\boldsymbol{\mathbf{d}})\mathcal{F}^{*}_{b}(\overline{k},\boldsymbol{\mathbf{d}})e^{i\left[\overline{k}\boldsymbol{\mathbf{d}}\cdot\boldsymbol{\mathbf{\delta\theta}}_{ba}\right]}\right\\}e^{\frac{-\sigma_{k}^{2}}{4}\left(\boldsymbol{\mathbf{d}}\cdot\boldsymbol{\mathbf{\delta\theta}}_{ba}\right)^{2}}\Bigg{\\}}.$

The suppression of the one-source terms
$\left|\mathcal{F}_{j}(\overline{k},\boldsymbol{\mathbf{d}})\right|^{2}\to 0$
as $kd\theta_{j}\to\infty$ in the first line of eq. 2.34 is how stellar
diameters can be determined. Not surprisingly, the finite source sizes also
suppress the two-source term in the second line. Note that
$\mathcal{F}_{a}\mathcal{F}_{b}^{*}$ is in general a complex number that can
give an additive contribution to the phase of interest, namely
$\overline{k}\boldsymbol{\mathbf{d}}\cdot\boldsymbol{\mathbf{\delta\theta}}_{ba}$.
For sources that are reflection-symmetric along the axis defined by
$\boldsymbol{\mathbf{d}}$ projected onto the celestial sphere, which is
guaranteed for _any_ baseline orientation as long as the source is spherically
symmetric, the form factor is real and does not contribute to the phase. Even
if $\mathcal{F}_{j}$ is not real, its phase can be modeled and estimated from
the one- and two-source intensity fringes. We assume that this is done at
sufficient fidelity and will ignore this phase hereafter.

As we will see in section 2.4, differential astrometry with intensity
interferometry is best performed on compact sources with high surface
brightness, so stars and quasars are the most suitable targets. Let us compute
the flux and form factor of eqs. 2.32 and 2.33 for these two types of objects.
For simplicity, we model stars as single-temperature disks

$\displaystyle
T_{\text{s}}(\hat{\boldsymbol{\mathbf{\theta}}})=T_{\text{s}}\,\Theta_{\text{H}}(\theta_{\text{s}}-\theta)$
(2.35)

where the temperature equals $T_{\text{s}}$ on the disk of angular radius
$\theta_{\text{s}}=R_{\text{s}}/D_{\text{s}}$, with $R_{\text{s}}$ and
$D_{\text{s}}$ the physical stellar radius and line-of-sight distance. This
gives a flux per spectral channel of

$\displaystyle I_{\text{s}}(\overline{k})=\frac{\hbar
c^{2}}{(2\pi)^{3/2}}\frac{\sigma_{k}\overline{k}^{3}\theta_{\text{s}}^{2}}{e^{\hbar
c\overline{k}/k_{\mathrm{B}}T_{\text{s}}-1}},$ (2.36)

and a simple form factor

$\displaystyle\mathcal{F}_{\text{s}}(\overline{k},\boldsymbol{\mathbf{d}})=\frac{2J_{1}(\overline{k}d\theta_{\text{s}})}{\overline{k}d\theta_{\text{s}}}$
(2.37)

which scales as
$-\sqrt{8/\pi}\cos(\overline{k}d\theta_{\text{s}}+\pi/4)/(\overline{k}d\theta_{\text{s}})^{3/2}$
at $\overline{k}d\theta_{\text{s}}\gg 1$.

For quasars, we assume the emission is also optically thick with a local
temperature function of:

$\displaystyle
T_{\mathrm{q}}(\hat{\boldsymbol{\mathbf{\theta}}})=\frac{T_{500}}{1+z_{\mathrm{q}}}\left(\frac{R_{500}}{R}\right)^{3/4}=T_{500}\left(\frac{\theta_{500}}{\theta}\right)^{3/4}$
(2.38)

where $T_{500}=28{,}776\,\mathrm{K}$ is the temperature corresponding to a
wavelength of $\lambda=500\,\mathrm{nm}$, and $R_{500}$ is the physical radius
(in the quasar’s local frame) at which this temperature is achieved. The
apparent angular radius
$\theta_{500}=R_{500}/[D_{\mathrm{q}}(1+z_{\mathrm{q}})^{4/3}]$ is where this
temperature is achieved in the _observer’s_ frame, after accounting for the
redshift of the quasar $z_{\mathrm{q}}$ and its angular diameter distance
$D_{\mathrm{q}}$. This model is a simplified version of the emission from a
face-on accretion disk in the model of ref. [70], with temperature function in
the quasar’s frame:

$\displaystyle
T_{\text{q}}(R)=T_{\mathrm{q},0}\left(\frac{R_{\text{isco}}}{R}\right)^{3/4}\left[1-\left(\frac{R_{\text{isco}}}{R}\right)^{1/2}\right]^{1/4},$
(2.39)

with $T_{\mathrm{q},0}$ is a reference temperature given by

$\displaystyle T_{\text{q},0}$
$\displaystyle=\left(\frac{3G^{2}M_{\text{BH}}^{2}m_{p}f_{\text{Edd}}}{2c\sigma_{\text{B}}\sigma_{\text{T}}\eta_{\text{q}}R_{\text{isco}}^{3}}\right)^{1/4}\approx
3.33\times
10^{5}\,\text{K}\left(\frac{f_{\text{Edd}}}{\eta_{\text{q}}}\right)^{1/4}\left(\frac{10^{9}\,M_{\odot}}{M_{\text{BH}}}\right)^{1/4}\left(\frac{GM_{\mathrm{BH}}/c^{2}}{R_{\text{isco}}}\right)^{1/4}$
(2.40)

with $M_{\text{BH}}$ the black hole mass, $m_{p}$ the proton mass,
$f_{\text{Edd}}$ the bolometric luminosity relative to the Eddington
luminosity, $\eta_{\text{q}}$ the bolometric luminosity relative to the
accretion rate,
$\sigma_{\text{B}}=(\pi^{2}/60)k_{\text{B}}^{4}/\hbar^{3}c^{2}$ the
Stefan–Boltzmann constant,
$\sigma_{\text{T}}=(8\pi/3)(\hbar^{2}/c)\alpha^{2}/m_{e}^{2}$ the Thomson
scattering cross-section, and $R_{\text{isco}}$ the innermost stable circular
orbit within the accretion disk (equaling $6GM_{\text{BH}}/c^{2}$ for a non-
spinning black hole, and $GM_{\text{BH}}/c^{2}$ for a maximally spinning black
hole). The apparent angular radius of the accretion disk is
$\theta_{\text{isco}}=R_{\text{isco}}/D_{\text{q}}$.

In the simplified model of eq. 2.38, the observed flux from the quasar per
spectral channel is then:

$\displaystyle I_{\mathrm{q}}(\overline{k})=\frac{\xi_{\mathrm{q}}\hbar
c^{2}}{(2\pi)^{3/2}}\,\sigma_{k}\overline{k}^{1/3}\,\left(\frac{k_{\mathrm{B}}T_{500}}{\hbar
c}\right)^{8/3}\,\theta_{\mathrm{500}}^{2},$ (2.41)

with $\xi_{\mathrm{q}}=\Gamma(11/3)\zeta(8/3)\approx 5.15$, and the form
factor is:

$\displaystyle\mathcal{F}_{\mathrm{q}}(\overline{k},\boldsymbol{\mathbf{d}})=\xi^{-1}_{\mathrm{q}}\int_{0}^{\infty}\mathrm{d}y\,\frac{2y}{\exp
y^{3/4}-1}J_{0}\left[\left(\frac{k_{\mathrm{B}}T_{500}}{\hbar
c}\right)^{4/3}\,\overline{k}^{-1/3}\,d\,\theta_{500}\,y\right].$ (2.42)

This form factor depends on $\overline{k}$ and $d$ through the combination
$a=(k_{\mathrm{B}}T_{500}/\hbar
c)^{4/3}\,\overline{k}^{-1/3}\,d\,\theta_{500}$; when $a\ll 1$, the form
factor goes to unity, while for $a\gg 1$, it is approximately
$\mathcal{F}_{\mathrm{q}}\approx 0.3a^{-5/4}$. This is a slightly less severe
suppression ($\propto d^{-5/4}$) than the form factor scaling for a star
($\propto d^{-3/2}$).

###  2.4 Noise and light-centroiding precision

In this section, we estimate the statistical noise on our primary
observable—the excess intensity correlation of eq. 2.21—allowing us to
calculate the signal-to-noise ratio (SNR) and light-centroiding precision for
different sources. We postpone a discussion of phase errors and other
systematic errors to section 3.

The data collected by each photodetector $p$ are arrival times $\\{t_{i}\\}$
for $N_{p}$ photons, from which one can build a simple estimator $\hat{I}_{p}$
(the hat indicates that it is a data-driven _estimate_) of the instantaneous
intensity at time $t_{p}$:

$\displaystyle\hat{I}_{p}(t_{p})=\frac{\hbar
c\overline{k}}{\eta_{p}A_{p}}\sum_{i=1}^{N_{p}}\frac{e^{\frac{-(t_{p}-t_{i})^{2}}{\sigma_{t}^{2}}}}{\sqrt{\pi}\sigma_{t}}.$
(2.43)

Above, $\overline{k}$ is the mean wavevector of the light (assumed to be
observed in a fractionally narrow band), $\eta_{p}$ is the photodetection
efficiency (excluding transmissivity and reflectivity factors from the
splitter and combiner in section 2.2), and $A_{p}$ is the telescope’s aperture
area. The effective timing precision _per photodetector_ is taken to be
$\sigma_{t}/\sqrt{2}$ with $\sigma_{t}$ from eq. 2.8, so that relative photon
arrival times, i.e. $t_{2}-t_{1}$, have relative timing precision
$\sigma_{t}$. An unbiased, optimal estimator of the excess fractional
intensity correlation between two detectors, time averaged over a time
interval $t_{\text{obs}}$, is then:

$\displaystyle\hat{C}(\boldsymbol{\mathbf{d}},\tau)=\frac{\overline{\hat{I}_{1}(t)\hat{I}_{2}(t+\tau)}}{\overline{\hat{I}_{1}(t)}~{}\overline{\hat{I}_{2}(t+\tau)}}-1.$
(2.44)

The overline signifies time-averaging according to:

$\displaystyle\overline{f(t)}\equiv\int_{t-t_{\text{obs}}/2}^{t+t_{\text{obs}}/2}\frac{\text{d}t^{\prime}}{t_{\text{obs}}}\,f(t^{\prime}),$
(2.45)

for any time-dependent observable $f(t)$. The expectation value of the
estimator in eq. 2.44 is $\langle\hat{C}\rangle=C$ with $C$ from eq. 2.21;
further details can be found in appendix B.1. (We assume that the time
dependence of $C$ during $t_{\text{obs}}$ is negligible.)

The estimator for the intensity correlation in eq. 2.44 only receives
contributions when photodetectors 1 and 2 each record photons with relative
times of arrival (given a fixed offset $\tau$) separated by
$\mathcal{O}(\sigma_{t})$ or less. Most photons detected at photodetector 1
will not have such a “simultaneous” photon at photodetector 2, and the excess
fractional correlation is small numerically (of order $C\sim
1/c\sigma_{k}\sigma_{t}\ll 1$), so obtaining a high SNR requires sufficiently
bright sources and long observation times, so that $N_{p}\gg 1$. In what
follows, we can therefore safely neglect Poisson fluctuations in the _total
number_ $N_{p}$ of photons, and only consider the arrival times $\\{t_{i}\\}$
as random variables. We further assume

$\displaystyle\frac{1}{c\overline{k}}\ll\frac{1}{c\sigma_{k}}\ll\sigma_{t}\ll
t_{\text{reset}}\ll\frac{t_{\mathrm{obs}}}{N_{p}},$ (2.46)

though modifications of the formulae below for weak or reversed hierarchies
are not difficult. Equation 2.46 is equivalent to assuming a strong hierarchy
of time scales, in ascending order: the period of light ($1/c\overline{k}$),
the coherence time of the light ($1/c\sigma_{k}$), the relative timing
precision ($\sigma_{t}$), the reset time of the photodetector
($t_{\text{reset}}$), and the typical time between photon counts
($t_{\text{obs}}/N_{p}$). With those assumptions, it is straightforward to
calculate the expected estimated intensity:

$\displaystyle\left<\overline{\hat{I}_{p}}\right>=\frac{\hbar
c\overline{k}}{\eta_{p}A_{p}}\frac{N_{p}}{t_{\text{obs}}}\equiv\langle
I_{p}\rangle,$ (2.47)

and, more tediously, the statistics-limited standard deviation on $\hat{C}$:

$\displaystyle\sigma_{\hat{C}}$
$\displaystyle\simeq\sqrt{\frac{\text{Var}\left\\{\overline{\hat{I}_{1}\hat{I}_{2}}\right\\}}{\langle
I_{1}\rangle\langle
I_{2}\rangle}}\simeq\sqrt{\frac{t_{\text{obs}}}{\sqrt{4\pi}\sigma_{t}}}\frac{1}{\sqrt{N_{1}N_{2}}}=\sqrt{\frac{1}{\sqrt{4\pi}\sigma_{t}t_{\text{obs}}}}\frac{\hbar
c\overline{k}}{\sqrt{\eta_{1}\eta_{2}A_{1}A_{2}}}\frac{1}{\sqrt{\langle
I_{1}\rangle\langle I_{2}\rangle}}.$ (2.48)

We provide derivations of eqs. 2.47 and 2.48 in appendix B.1.

Figure 6: Single-channel precision $\sigma_{\hat{C}}$ from eq. 2.48 on the
excess intensity correlation, as a function of wavenumber $\overline{k}$ (for
reference, the wavelength range of
$\overline{\lambda}=[300\,\text{nm},1\,\mathrm{\mu m}]$ is indicated by black,
dashed, vertical lines). The light gray, yellow, red, and blue curves
correspond to a white dwarf (“WD”, $T=20{,}000\,\text{K}$, $D=10\,\text{pc}$,
$R=10^{-2}\,R_{\odot}$), Sun-like main sequence star (“MS”,
$T=6{,}000\,\text{K}$, $D=100\,\text{pc}$, $R=1\,R_{\odot}$), red giant (“RG”,
$T=4{,}000\,\text{K}$, $D=5\,\text{kpc}$, $R=10^{2}\,R_{\odot}$), and blue
giant (“BG”, $T=20{,}000\,\text{K}$, $D=5\,\text{kpc}$, $R=10\,R_{\odot}$),
respectively. The pink curve shows $\sigma_{\hat{C}}$ for a quasar at redshift
$z_{\mathrm{q}}=1$ with optical radius $R_{500}=10^{16}\,\mathrm{cm}$ (eq.
2.38). We assume telescope parameters from Phase II in table 1, and an
integration time $t_{\text{obs}}=10^{4}\,\text{s}$. The solid black line
indicates the single-source excess fractional correlation (eq. 2.11). The
integrated SNR of eq. 2.49 between $1\,\mathrm{\mu m}$ and $300\,\text{nm}$ is
$\\{93,432,371,91,2.2\\}$ for these sources
$\\{\text{WD},\text{MS},\text{RG},\text{BG},\text{quasar}\\}$.

We can estimate $\sigma_{\hat{C}}$ numerically by using the flux-per-channel
formula from eq. 2.32. If one has $n_{p}^{\text{arr}}$ telescopes at site
$p=1,2$, the noise is reduced by a factor of
$1/\sqrt{n_{1}^{\text{arr}}n_{2}^{\text{arr}}}$ by averaging at each of the
two sites. Finally, we correct for the suppression due to the two uncorrelated
polarizations of light from unpolarized sources by multiplying by $1/2$ the
total signal-to-noise ratio (SNR) on the excess fractional intensity
correlation:

$\displaystyle\text{SNR}=\frac{1}{2}\sqrt{{n_{1}^{\text{arr}}n_{2}^{\text{arr}}}\sum_{\overline{k}}\left(\frac{C(\boldsymbol{\mathbf{d}},\tau)}{\sigma_{\hat{C}}}\right)^{2}_{\overline{k}}},$
(2.49)

where the sum is over channels with different $\overline{k}$ separated from
each other by more than $\sigma_{k}$, e.g. with a logarithmic spectral spacing
between channels of the form $e^{2n\sigma_{k}/\overline{k}}$ for integer $n$.
In figure 6, we plot $\sigma_{\hat{C}}$ from eq. 2.48 for a single pair of
telescopes ($n_{p}^{\text{arr}}=1$) as a function of $\overline{k}$ for
thermal spectra of four different reference stars, convoluted with a narrow
spectral filter of width $\sigma_{k}$. For the parameters plotted in figure 6,
the integrated signal-to-noise ratio over $6{,}020$ logarithmically spaced
channels between $\overline{\lambda}=1\,\mu\text{m}$ and
$\overline{\lambda}=300\,\text{nm}$ from eq. 2.49 is
$\text{SNR}=\\{93,432,371,91\\}$ for the white dwarf, main sequence, red
giant, and blue giant stars, respectively. Likewise, for the fiducial quasar
shown in pink in figure 6, the integrated SNR equals $2.2$, despite
$\sigma_{\hat{C}}>C\simeq 1/2\sqrt{2}c\sigma_{k}\sigma_{t}$ in any one
spectral channel.

For differential astrometry using EPIC, the relevant term in $C$ is reduced by
a factor of $1/4$ (cfr. eq. 2.28), while the observed fluxes $\propto\langle
I_{p}\rangle$ are reduced by a factor of $1/2$ per source if
$\epsilon=1/\sqrt{2}$ (cfr. eq. 2.22), reducing the total SNR by a factor of
$1/8$ (compared to not having any path extension at all). We note though that
compared to figure 6, this flux loss is compensated by observing _two_ sources
simultaneously. In addition, with the use of a polarizing beamsplitter, one
could theoretically recover a factor of $\sqrt{2}$ in SNR and light-
centroiding precision [39, 71]. Likewise, one could recover an extra factor of
2 from the second output port of the beam recombiner (by also recording light
going to the right from the last semi-transparent plate in figure 5(b)).

The main application of EPIC is to measure the separation—and especially their
change over time—between the light centroids of two point-like sources. This
can be achieved at a precision better than $\sigma_{\theta_{\text{res}}}$ from
eq. 2.9 if the differential astrometry fringe of eq. 2.28 is detected at high
SNR. Denote by $\sigma_{\delta\theta}$ the precision to which the light-
centroid separation
$\boldsymbol{\mathbf{\theta}}_{ba}\cdot\hat{\boldsymbol{\mathbf{d}}}$ along
the baseline direction can be measured. A standard error propagation
calculation on eq. 2.34 then yields a light-centroiding precision of

$\displaystyle\sigma_{\delta\theta}$
$\displaystyle=2\left[n_{1}^{\text{arr}}n_{2}^{\text{arr}}\sum_{\overline{k}}\left(\frac{\sigma_{\theta_{\text{res}}}\sigma_{\hat{C}}}{\left|\sin(\overline{k}\boldsymbol{\mathbf{d}}\cdot\boldsymbol{\mathbf{\delta\theta}}_{ba})\right|}\frac{2\sqrt{2}c\sigma_{k}\sigma_{t}}{\widetilde{I}_{a}\widetilde{I}_{b}\left|\mathcal{F}_{a}\mathcal{F}^{*}_{b}\right|}\right)^{-2}\right]^{-1/2}$
(2.50)

when combining all channels labeled by $\overline{k}$, between
$k_{\text{min}}<\overline{k}<k_{\text{max}}$. In the limiting case of two
identical detectors ($n^{\mathrm{arr}}_{1}=n^{\mathrm{arr}}_{2}=1$,
$\eta_{1}=\eta_{2}=\eta$, $A_{1}=A_{2}=A$, etc.), fixed spectral resolution
($\overline{k}/\sigma_{k}$ independent of $\overline{k}$), and identical,
unpolarized, sources $a$ and $b$, we find:

$\displaystyle\sigma_{\delta\theta}\simeq\frac{2^{13/2}\pi^{5/4}\hbar^{3}c^{3}}{k_{\mathrm{B}}^{3}}\frac{1}{\eta
Ad_{\parallel}}\sqrt{\frac{\sigma_{t}}{t_{\text{obs}}}}\sqrt{\frac{\sigma_{k}}{\overline{k}}}\begin{cases}\frac{1}{T_{\text{s}}^{3}\theta_{\text{s}}^{2}}\left[\int_{x_{\mathrm{min,s}}}^{x_{\mathrm{max,s}}}\text{d}x\,\frac{x^{5}}{(e^{x}-1)^{2}}\mathcal{F}^{4}_{\text{s}}(xk_{\mathrm{B}}T_{\text{s}}/\hbar
c,\boldsymbol{\mathbf{d}})\right]^{-1/2}\\\
\frac{1}{T_{500}^{3}\theta_{500}^{2}}\left[\int_{x_{\mathrm{min,500}}}^{x_{\mathrm{max,500}}}\text{d}x\,x^{-1/3}\mathcal{F}^{4}_{\text{q}}(xk_{\mathrm{B}}T_{500}/\hbar
c,\boldsymbol{\mathbf{d}})\right]^{-1/2},\end{cases}$ (2.51)

where $x_{\mathrm{min,s}}\equiv\hbar
ck_{\text{min}}/k_{\mathrm{B}}T_{\text{s}}$ and
$x_{\mathrm{min,500}}\equiv\hbar
ck_{\text{min}}/k_{\mathrm{B}}T_{\text{500}}$, and similarly for
$x_{\mathrm{max,s}}$ and $x_{\mathrm{max,500}}$. To get to eq. 2.51 from eq.
2.50, we approximated
$\sin^{2}(\overline{k}\boldsymbol{\mathbf{d}}\cdot\boldsymbol{\mathbf{\delta\theta}}_{ba})\sim
1/2$, and turned the sum over $\overline{k}$ into an integral according to
$\sum_{\overline{k}}\to\int\text{d}\overline{k}/2\sigma_{k}$. The functions
$\mathcal{F}_{\text{s}}$ and $\mathcal{F}_{\text{q}}$ are defined in eqs. 2.37
and 2.42, respectively. The effective baseline $d_{\parallel}$ is the
magnitude of the baseline vector component of $\boldsymbol{\mathbf{d}}$
parallel to the source separation $\boldsymbol{\mathbf{\theta}}_{ba}$.

Figure 7: Light-centroiding precision $\sigma_{\delta\theta}$ from eq. 2.51 as
a function of baseline distance $d$ (assuming optimal alignment, i.e.
$\boldsymbol{\mathbf{d}}\cdot\boldsymbol{\mathbf{\theta}}_{ba}=d\theta_{ba}$)
between _identical_ stars and quasars, indicated by color and contours (in
units of $\mu\mathrm{as}$). The parameters for the telescopes (Phase II in
table 1), stars, and quasars are the same as in figure 6.

The factors outside the curly brackets in eq. 2.51 are in the observer’s
control: higher efficiencies $\eta$, larger collecting areas $A$, better
timing precision $\sigma_{t}$, longer observation times $t_{\text{obs}}$, and
higher spectral resolution $\overline{k}/\sigma_{k}$ generally improve the
light-centroiding precision. Inside the curly brackets, we have provided the
expression relevant for astrometry between two identical stars (top line) of
temperature $T_{\text{s}}$ and angular radius $\theta_{\text{s}}$, and two
identical quasar images (bottom line) with apparent angular optical radius of
$\theta_{500}$.cLight centroiding precision improves for higher-temperature
sources (higher $T_{\text{s}}$), and for sources that subtend larger angles on
the sky (larger $\theta_{\text{s}}$ in the case of stars, or $\theta_{500}$
for quasars).

In figure 7, we plot the light-centroiding precision of eq. 2.51 for the same
4 stars and fiducial quasar, and instrumental parameters as in figure 6; e.g.
the gray curve is the differential astrometric light-centroiding precision
between two identical white dwarfs of $T_{\text{s}}=10^{4}\,\text{K}$ with
$R_{\text{s}}=10^{-2}\,R_{\odot}$ at $D_{\text{s}}=10\,\text{pc}$. We can see
that the astrometric error scales as $\sigma_{\delta\theta}\propto
1/d_{\parallel}$ until the sources are resolved, so there is an optimum
baseline distance $d_{\text{opt}}$. For quasars, the optimum baseline distance
is just a simple linear function of $\theta_{500}$:

$\displaystyle d_{\text{opt,q}}\approx
2.31\,\mathrm{km}\,\left(\frac{10^{-11}\,\mathrm{rad}}{\theta_{500}}\right)$
(2.52)

where we assumed $k_{\mathrm{min}}=2\pi/(1000\,\mathrm{nm})$ and
$k_{\mathrm{max}}=2\pi/(300\,\mathrm{nm})$. In figure 8, we display the light-
centroiding precision at this optimal baseline distance $d_{\text{opt}}$,
assuming $d=d_{\parallel}$ for stellar sources as a function of
$\theta_{\text{s}}$ and $T_{\text{s}}$. We show the equivalent quantity in
figure 19 for quasars as a function of $\theta_{500}$, assuming the optimum
baseline of eq. 2.52.

Figure 8: Light-centroiding precision $\sigma_{\delta\theta}$ from eq. 2.51
between two identical stars of angular size
$\theta_{\text{s}}=R_{\text{s}}/D_{\text{s}}$ and temperature $T_{\text{s}}$,
at the optimal baseline distance $d^{\text{opt}}$ for a Phase II EPIC (see
table 1).

###  2.5 Atmospheric phases

In this section, we outline the effects of non-negligible propagation phases
on EPIC at “large” angles. We compute the size of the atmospheric aberration
and its resulting limitation on the field of view of EPIC.

Let us start with the same electric field as in eqs. 2.2 and 2.18, but retain
the phases $\widetilde{\phi}^{(p)}_{i\alpha}(t_{p})$ _not_ intrinsic to the
source but accumulated along the path from the source $i$ to the detector $p$
for a mode of frequency $k_{\alpha}$:

$\displaystyle E_{p}(t_{p})$
$\displaystyle=\epsilon\sqrt{1-\epsilon^{2}}\sum_{i=1}^{N_{\theta}}\sum_{\alpha=1}^{N_{k}}\sum_{n=0}^{1}e^{-ink_{\alpha}\ell_{p}}E_{i\alpha}\exp\left\\{i\left[k_{\alpha}(ct_{p}-r_{ip})+\phi^{\text{em}}_{i\alpha}+\widetilde{\phi}^{(p)}_{i\alpha}(t_{p})\right]\right\\},$
(2.53)

In the continuum limit, the relevant extended-path intensity correlator is
modified to:

$\displaystyle
C(\boldsymbol{\mathbf{d}},\tau)=\frac{\epsilon^{4}\left(1-\epsilon^{2}\right)^{2}}{\left<I_{1}\right>\left<I_{2}\right>}\int\frac{\text{d}^{2}b_{1}}{A_{1}}\int\frac{\text{d}^{2}b_{2}}{A_{2}}\int\text{d}(\Delta
t)\frac{1}{\sqrt{2\pi}\sigma_{t}}e^{\frac{-(\Delta t)^{2}}{2\sigma_{t}^{2}}}$
(2.54)
$\displaystyle\times\left|\int\text{d}k\int\text{d}\Omega\,\frac{\text{d}I}{\text{d}k\,\text{d}\Omega}\big{(}1+e^{-ik\ell_{1}}\big{)}\big{(}1+e^{ik\ell_{2}}\big{)}e^{i\left[k\left(c(\Delta
t-\tau)-\hat{\boldsymbol{\mathbf{\theta}}}\cdot(\boldsymbol{\mathbf{d}}+\boldsymbol{\mathbf{\Delta
b}})\right)+\widetilde{\phi}^{(1)}(t,\hat{\boldsymbol{\mathbf{\theta}}},k)-\widetilde{\phi}^{(2)}(t+\tau,\hat{\boldsymbol{\mathbf{\theta}}},k)\right]}\right|^{2},$

which should be regarded as the generalization of eq. 2.21 to account for
propagation phases. The limit towards conventional intensity interferometry is
attained in the limit $\ell_{p}\to 0$. The expected intensities $\langle
I_{p}\rangle$ are unchanged from eq. 2.19.

For two point-like sources $a$ and $b$, we find

$\displaystyle
C^{\text{opt}}(\boldsymbol{\mathbf{d}})\simeq\frac{1}{4\sqrt{2}c\sigma_{k}\sigma_{t}}$
$\displaystyle\Bigg{\\{}\widetilde{I}_{a}^{2}+\widetilde{I}_{b}^{2}+2\widetilde{I}_{a}\widetilde{I}_{b}\cos\left[\overline{k}\boldsymbol{\mathbf{d}}\cdot\boldsymbol{\mathbf{\delta\theta}}_{ba}+\Delta\widetilde{\phi}^{(1,2)}(\overline{k},\hat{\boldsymbol{\mathbf{\theta}}}_{a},\hat{\boldsymbol{\mathbf{\theta}}}_{b})\right]e^{\frac{-\sigma_{k}^{2}}{4}\left(\boldsymbol{\mathbf{d}}\cdot\boldsymbol{\mathbf{\delta\theta}}_{ba}\right)^{2}}\Bigg{\\}}$
(2.55)

as the generalization of 2.28 for example. In the above, we have defined the
doubly-differential noise phase:

$\displaystyle\Delta\widetilde{\phi}^{(1,2)}(\overline{k},\hat{\boldsymbol{\mathbf{\theta}}}_{a},\hat{\boldsymbol{\mathbf{\theta}}}_{b})\equiv\widetilde{\phi}^{(1)}(\overline{k},\hat{\boldsymbol{\mathbf{\theta}}}_{a})+\widetilde{\phi}^{(2)}(\overline{k},\hat{\boldsymbol{\mathbf{\theta}}}_{b})-\widetilde{\phi}^{(1)}(\overline{k},\hat{\boldsymbol{\mathbf{\theta}}}_{b})-\widetilde{\phi}^{(2)}(\overline{k},\hat{\boldsymbol{\mathbf{\theta}}}_{a}),$
(2.56)

whose time dependence we have implicitly suppressed (since the relevant
quantity is its time average).

We are particularly interested in calculating the atmospheric contribution to
the doubly-differential phase of eq. 2.56 due to the fluctuating index of
refraction along the light paths. Denote by
$n[\overline{k},\boldsymbol{\mathbf{x}}_{ip}(s)]$ the deviation of the index
of refraction from its mean at wavenumber $\overline{k}$ along the path from
detector $p$ to source $i$:
$\boldsymbol{\mathbf{x}}_{ip}(s)\equiv\boldsymbol{\mathbf{r}}_{p}+\hat{\boldsymbol{\mathbf{\theta}}}_{i}s$
with $0<s<r_{ip}$. The total refractive phase accumulation along the path is:

$\displaystyle\widetilde{\phi}^{(p)}(\overline{k},\hat{\boldsymbol{\mathbf{\theta}}}_{i})=\overline{k}\int_{0}^{r_{ip}}\text{d}s\,n[\overline{k},\boldsymbol{\mathbf{x}}_{ip}(s)].$
(2.57)

We have explicitly subtracted out the mean index of refraction, so $\langle
n[\overline{k},\boldsymbol{\mathbf{x}}(s)]\rangle_{\mathrm{atm}}$ vanishes by
construction along any path $\boldsymbol{\mathbf{x}}(s)$ through the
atmosphere, and thus also the corresponding
$\langle\widetilde{\phi}\rangle_{\mathrm{atm}}$. This subtraction of the mean
can be done to a sufficient fidelity, down to an effective precision less than
a millimeter, corresponding to $\lesssim 3\,\mathrm{ps}$ delay [72, 73, 74].

However, there are _fluctuations_ in $n$, as measured by the structure
function [75, 76, 77, 78]:

$\displaystyle
D_{n}(r)\equiv\left\langle(n[\boldsymbol{\mathbf{r}}^{\prime}+\boldsymbol{\mathbf{r}}]-n[\boldsymbol{\mathbf{r}}^{\prime}])^{2}\right\rangle_{\mathrm{atm}}=C_{n}^{2}(h)r^{2/3}\quad\text{for
}l_{0}\ll r\ll L_{0},$ (2.58)

where we have dropped the dependence on $\overline{k}$ for notational
simplicity. The structure function does not depend on the reference position
$\boldsymbol{\mathbf{r}}^{\prime}$ except through the slow dependence on
altitude $h$ in the atmosphere. The two-thirds scaling follows from
Kolmogorov’s ansatz for turbulence, and is an _overestimate_ both above the
outer scale $L_{0}\sim 10^{1\pm 1}\,\mathrm{m}$ (where $D_{n}(r)$ asymptotes
to a constant $\langle n^{2}\rangle$) and below the inner scale $l_{0}\sim
10^{-2.5\pm 0.5}\,\mathrm{m}$ (where $D_{n}(r)\approx r^{2}$). The constant
$C_{n}$ has dimensions of $\mathrm{[length]}^{-1/3}$, depends on the weather,
and typically decreases with altitude. Typical values are $C_{n}\sim
10^{-8}\,\mathrm{m}^{-1/3}$ for altitudes between $1$–$10\,\mathrm{km}$, as
depicted in figure 20 for the SLC night model [79]. Also useful is the
correlation function

$\displaystyle B_{n}(r)\equiv\left\langle
n[\boldsymbol{\mathbf{r}}^{\prime}+\boldsymbol{\mathbf{r}}]n[\boldsymbol{\mathbf{r}}^{\prime}]\right\rangle_{\mathrm{atm}}.$
(2.59)

The structure function can be expressed in terms of the correlation function
as $D_{n}(r)=2[B_{n}(0)-B_{n}(r)]$.

Figure 9: Isoplanatic patch angle $\theta_{0}$ (black solid lines) as a
function of wavenumber $\overline{k}$ for three different zenith angles
$\gamma=\\{0^{\circ},45^{\circ},60^{\circ}\\}$. Atmospheric refraction
suppresses the intensity fringe for differential astrometry at an angle
$\theta_{ba}$ as $\exp\\{-(\theta_{ba}/\theta_{0})^{5/3}\\}$ (cfr. eq. 2.62).
The blue dotted line indicates the dynamic range $\sigma_{\Delta\theta}$ of
eq. 2.15 for intensity interferometry without path extensions at
$d=100\,\text{km}$ and $\overline{k}/\sigma_{k}=5{,}000$. The effective field
of view (FOV) for traditional intensity interferometry is
$\sigma_{\Delta\theta}$, while it is as large as $\theta_{0}$ for EPIC, an
increase by more than three orders of magnitude at the chosen parameters. We
plot the angular size $\theta_{\text{seeing}}$ of the seeing disk from eq.
2.63 as green dash-dotted lines
($\gamma=\\{60^{\circ},45^{\circ},0^{\circ}\\}$) above this line, two sources
can be resolved from a ground-based observatory at an excellent site without
adaptive optics. The brown dashed line indicates the critical angle
$\theta_{\mathrm{ap}}$ above which wavefront errors need to be taken into
account for a $4\,\text{m}$ aperture diameter (the
$\boldsymbol{\mathbf{b}}_{p}$ dependence in eq. 2.55).

The doubly differential atmospheric refraction phase from eq. 2.56 is a
(rapidly) fluctuating quantity with expected standard deviation:

$\displaystyle\sigma_{\phi}^{\text{atm}}\equiv\sqrt{\Big{\langle}\left[\Delta\widetilde{\phi}^{(1,2)}(\overline{k},\hat{\boldsymbol{\mathbf{\theta}}}_{a},\hat{\boldsymbol{\mathbf{\theta}}}_{b})\right]^{2}\Big{\rangle}_{\mathrm{atm}}}\simeq\sqrt{2}\left(\frac{\theta_{ba}}{\theta_{0}(\overline{k},\gamma)}\right)^{5/6},$
(2.60)

where $\gamma$ is the angle of the sources away from zenith (assumed to be the
same for both detector locations, a reasonable assumption for baselines
smaller than Earth’s radius $d\ll R_{\oplus}$), and $\theta_{0}$ is the
_isoplanatic patch angle_ [80, 81]:

$\displaystyle\theta_{0}(\overline{k},\gamma)\equiv\left[2.9\overline{k}^{2}\int\mathrm{d}s\,C_{n}^{2}(s\cos\gamma)s^{5/3}\right]^{-3/5}\approx
2.7\,\mathrm{arcsec}\left(\frac{\overline{\lambda}}{500\,\text{nm}}\right)^{6/5}\cos^{5/8}(\gamma).$
(2.61)

In the latter numerical estimate, we have used the SLC night model. We derive
eq. 2.60 in appendix B.2. This fluctuating atmospheric phase suppresses the
fringe of differential astrometry as
$\int\text{d}(\Delta\widetilde{\phi})(\sqrt{2\pi}\sigma_{\phi}^{\text{atm}})^{-1}e^{-i\Delta\widetilde{\phi}^{2}/2(\sigma_{\phi}^{\text{atm}})^{2}}e^{-i\Delta\widetilde{\phi}}=e^{-(\sigma_{\phi}^{\text{atm}})^{2}/2}$,
which when applied to eq. 2.28 gives:

$\displaystyle
C^{\text{opt}}(\boldsymbol{\mathbf{d}})\simeq\frac{1}{4\sqrt{2}c\sigma_{k}\sigma_{t}}$
$\displaystyle\Bigg{\\{}\widetilde{I}_{a}^{2}+\widetilde{I}_{b}^{2}+2\widetilde{I}_{a}\widetilde{I}_{b}\cos\left[\overline{k}\boldsymbol{\mathbf{d}}\cdot\boldsymbol{\mathbf{\delta\theta}}_{ba}\right]\exp\left\\{-\left[\frac{\theta_{ba}}{\theta_{0}(\overline{k},\gamma)}\right]^{5/3}\right\\}e^{\frac{-\sigma_{k}^{2}}{4}\left(\boldsymbol{\mathbf{d}}\cdot\boldsymbol{\mathbf{\delta\theta}}_{ba}\right)^{2}}\Bigg{\\}}.$
(2.62)

with any other contributions to
$\Delta\widetilde{\phi}^{(1,2)}(\overline{k},\hat{\boldsymbol{\mathbf{\theta}}}_{a},\hat{\boldsymbol{\mathbf{\theta}}}_{b})$
other than from the atmosphere ignored. Thus, any source separation smaller
than the isoplanatic patch angle, i.e.
$\theta_{ba}\lesssim\theta_{0}(\overline{k},\gamma)$, can be considered
acceptable.

The atmospheric phase error $\sigma_{\phi}^{\mathrm{atm}}$ for intensity
interferometry is dominated by atmospheric turbulence in the _upper_
atmosphere, as the quantity $h^{8/3}C_{n}^{2}(h)$ typically peaks near $h\sim
10\,\mathrm{km}$. Therefore, the altitude of the telescopes plays only a minor
role, and the observations can be taken near sea level, even in comparatively
poor atmospheric conditions. This is to be contrasted with the size of the
seeing disk:

$\displaystyle\theta_{\text{seeing}}(\overline{k},\gamma)=0.92\,\overline{k}^{1/5}\left[\int\mathrm{d}s\,C_{n}^{2}(s\cos\gamma)\right]^{3/5}\approx\frac{0.29\,\text{arcsec}}{\cos^{3/5}\gamma}\left(\frac{500\,\text{nm}}{\overline{\lambda}}\right)^{1/5},$
(2.63)

where the SLC night model was used again to obtain the (optimistic) numerical
estimate. At excellent observing sites, the seeing is smaller than the
isoplanatic angle, so suitable sources can be resolved from the ground with
some idea of their separation before EPIC is performed.

##  3 Observatory Design

In this section, we discuss the technical aspects of the extended-path
intensity correlation technique. We begin by outlining the telescope design
and array layout (section 3.1), and sketch out the operational protocol of an
EPIC observation (section 3.2). We focus on two technological advances on the
original intensity interferometry setup that allow us to perform intensity
interferometry (including EPIC) on fainter sources than previously possible:
fast photodetectors and readout electronics (section 3.3) and spectroscopic
splitting (section 3.4). The required tolerances on the optically active
surfaces and timing are summarized in section 3.5; a detailed conceptual
design report is left for future work.

###  3.1 Overview

| $D$ | $\sigma_{t}$ | $\mathcal{R}$ | $n_{\mathrm{arr}}$ | $\sigma_{\delta\theta}\,[\mathrm{\mu as}]$ | $\sigma_{\hat{\boldsymbol{\mathbf{\theta}}}}\,[\mathrm{arcsec}]$ | $\sigma_{\Delta\theta}\,[\mathrm{arcsec}]$
---|---|---|---|---|---|---|---
Phase I | $4\,\mathrm{m}$ | $30\,\mathrm{ps}$ | $5{,}000$ | 1 | $22.3$ | $5.24$ | $0.164$
Phase II | $10\,\mathrm{m}$ | $10\,\mathrm{ps}$ | $10{,}000$ | 1 | $1.46$ | $1.75$ | $0.327$
Phase III | $10\,\mathrm{m}$ | $3\,\mathrm{ps}$ | $20{,}000$ | 10 | $0.0564$ | $0.524$ | $0.656$
Table 1:  Fiducial EPIC array parameters in the near, medium, and long term
(Phases I, II, and III): aperture diameter $D$, relative timing precision
$\sigma_{t}$, spectral resolution $\mathcal{R}$, and number $n_{\mathrm{arr}}$
of telescope pairs sharing the same baseline. We assume a photodetection
efficiency of $\eta=0.5$ on top of the beamsplitters’
reflectivity/transmission coefficients. We quantify their performance by the
optimum light-centroiding precision $\sigma_{\delta\theta}$ after an
integration time $t_{\mathrm{obs}}=10^{4}\,\mathrm{s}$ for a pair of Sun-like
stars at a distance of $100\,\mathrm{pc}$ as well as the characteristic global
astrometric precision $\sigma_{\hat{\boldsymbol{\mathbf{\theta}}}}$ and
angular dynamic range $\sigma_{\Delta\theta}$ (at the optimum centroiding
baseline). For such source pair, the optimum (projected) baseline distance is
$\hat{\boldsymbol{\mathbf{\theta}}}_{ba}\cdot\boldsymbol{\mathbf{d}}\approx
0.71\,\mathrm{km}$, corresponding to a fiducial angular resolution of
$\sigma_{\theta_{\mathrm{res}}}\approx 23\,\mathrm{\mu as}$ at
$\overline{\lambda}=500\,\mathrm{nm}$ for all three EPIC Phases.

#### Telescopes

Each telescope within our proposed intensity interferometer array is
essentially a stellar spectrograph with moderately high spectral resolution,
an “internal amplitude interferometer” to create the path extension, and
relatively loose optical tolerances (see section 3.5 below). Due to the high
photon statistics required to tease out the weak intensity correlation, each
telescope should have a large aperture and be equipped with ultra-fast
photodetection electronics. The telescope design is illustrated in figure 5.
The fiducial photodetection efficiency is taken to be $\eta=0.5$, while other
reference specifications—the aperture diameter $D$, timing resolution
$\sigma_{t}$, spectral resolution $\mathcal{R}$, and number $n_{\mathrm{arr}}$
of telescopes per array site—are shown in table 1. No gains are expected from
a spectral resolution higher than $\sqrt{2}c\overline{k}\sigma_{t}\approx
5\times 10^{4}$ at $\overline{k}=2\pi/(500\,\mathrm{nm})$ and
$\sigma_{t}=10\,\mathrm{ps}$, cfr. eq. 2.11, which corresponds to a timing
precision sufficiently fast to fully resolve the coherence time of the light
in the spectral channel centered at a wavelength of $500\,\mathrm{nm}$. Larger
apertures are not inconceivable, especially given the low to moderate
tolerances required on the most of the optical surfaces, in particular the
primary mirror. Large steerable telescopes with aperture diameters of
$10$–$30\,\mathrm{m}$ are already in operation or construction for Cherenkov
cosmic ray imaging [43, 44, 45, 46, 47].

#### Variable path extension

The main conceptual difference between an ordinary spectrograph and a
constituent telescope of an EPIC array is that the latter requires an internal
beamsplitter and recombiner (with unequal path lengths). In light of eqs. 2.25
and 2.26, to cover all possible source separations $\theta_{ba}$ up to the
isoplanatic patch angle $\theta_{0}$ of eq. 2.61, the internal interferometer
of each telescope $p$ should have the capability to create a differential
optical path length of

$\displaystyle\ell_{p}\sim\frac{\theta_{ba}d}{2}\lesssim\frac{\theta_{0}d}{2}\sim
5\,\mathrm{cm}\left(\frac{\theta_{0}}{2\,\mathrm{arcsec}}\right)\left(\frac{d}{10\,\mathrm{km}}\right).$
(3.1)

This can be achieved by a corresponding tuning of the width $w_{p}$ of the
internal interferometer (at fixed $\gamma$ and $\gamma^{\prime}$), cfr. figure
5 and eq. 2.17. The precision to which $\ell_{p}$ is _adjustable_ should be on
the order of a wavelength, so that the intensity interferometer can operate
near the main (ghost) fringe, to adjust
$\boldsymbol{\mathbf{\theta}}^{\mathrm{ref}}_{ba}$ such that
$\boldsymbol{\mathbf{\delta\theta}}_{ba}\cdot\hat{\boldsymbol{\mathbf{d}}}\sim\sigma_{\theta_{\mathrm{res}}}$,
and to correct for aperture smearing, i.e. the slightly different angle of
incidence between the wavefronts from each source (the
$\boldsymbol{\mathbf{b}}$-dependence in eqs. 2.25 and 2.26). (In fact, as long
as the aperture smearing effects are corrected to sub-wavelength precision,
$|\boldsymbol{\mathbf{\delta\theta}}_{ba}\cdot\boldsymbol{\mathbf{d}}|\lesssim\sigma_{\Delta\theta}$
is also acceptable.) Aperture smearing is negated by taking
$\gamma^{\prime}-\gamma$ equal to $\theta_{ba}f/2f^{\prime}$ and with a
relative tilt in the same direction as
$\hat{\boldsymbol{\mathbf{\theta}}}_{ba}$ (see figure 5 and appendix C.2).

The positioning and control of the path extension require sub-wavelength
tolerances over tens of centimeters, which are mild compared to what has been
demonstrated in laboratory and observatory settings [82, 83]. For example, in
the CHARA interferometer, a laser metrology unit controls the mirrors to a
root-mean-squared error of $10$–$20\,\mathrm{nm}$ over distances of tens of
meters [26], while mirrors in adaptive optics systems achieve similar or
better control over smaller distances and angles [84, 85].

#### Array configuration

In this work, we primarily consider the case of a single pair of telescopes
separated by a distance $\boldsymbol{\mathbf{d}}$. Interesting science can be
obtained from a single pair of telescopes (section 4), but the potential of an
intensity interferometry array can grow quadratically with the number of
telescopes in the array. This quadratic scaling can be made precise: the
signal-to-noise ratio on the excess intensity correlation scales linearly with
the number of telescope pairs $n^{\mathrm{arr}}$ sharing the same baseline in
the array (eq. 2.49), so for the same light-centroiding precision
$\sigma_{\delta\theta}$ (eq. 2.50), the required integration time
$t_{\mathrm{obs}}$ scales as $1/(n^{\mathrm{arr}})^{2}$. Equivalently, the
light-centroiding precision can thus improve as $\sigma_{\delta\theta}\propto
1/n_{\mathrm{arr}}$ for fixed integration time.

For a fixed number of telescopes, the optimal layout of an EPIC array is
determined by the interplay of several considerations [86]. First, the optimal
projected baseline separation falls in the $1\,\mathrm{km}$–$100\,\mathrm{km}$
range, primarily depending on the angular size of the source (and more weakly
on source temperature). Second, the array should ideally have dense coverage
in the $(u,v)$ plane for faithful “image” reconstruction of the emission
region(s). Third, the array should have sensitivity to many (pairs of) sources
of varying angular sizes and locations on the sky. Fourth, the SNR should be
maximized.

The first three considerations dictate that, for a source at zenith, the
telescopes should be distributed along the boundary of (a slightly perturbed)
Reuleaux triangle [87], a curved triangle of constant width (and the constant-
width curve with the least rotational symmetry), with an overall size
determined by the maximum desired baseline. This configuration creates the
most uniform sampling (exact uniformity being impossible [88]) in a circular
disk in Fourier space, suitable for image reconstruction [89] and for
detecting sources of varying angular sizes. Because such a telescope
arrangement has a near-isotropic response, it is a good compromise for a
source not exactly at zenith, and for Earth rotation synthesis and lower-
elevation observations (third consideration). Slight, random perturbations
from the perfectly symmetric Reuleaux triangle shape further break rotational
symmetry, though ring- or ellipse-like configurations may be more desirable
for low-declination observations [90].

However, the fourth consideration favors some built-in degeneracy for an EPIC
array layout (e.g. in a square lattice or Y-shaped configuration) due to the
aforementioned $n_{\mathrm{arr}}$ scaling of the signal-to-noise ratio for
degenerate baselines. Furthermore, studies for optimal interferometric
configurations have focused largely on the case of narrow-bandwidth radio
interferometers; an EPIC array with good spectroscopic capabilities would have
a broadband response, i.e. any _single_ baseline would densely cover a “radial
spoke” in Fourier space over an octave or more, which may affect the optimal
array layout, and is an interesting topic for future study.

###  3.2 Observational Procedure

In sections 2.4 and 4, we implicitly assume that the source separation vector
$\boldsymbol{\mathbf{\theta}}_{ba}$ is already roughly known, and then study
the ultimate light-centroiding capabilities and applications of a pair of EPIC
telescopes. In practice, however, $\boldsymbol{\mathbf{\theta}}_{ba}$ may only
be predetermined to some precision $\sigma_{\delta\theta}^{(0)}$ from prior
observations. If this precision is of order the intrinsic angular resolution
(or better), i.e.
$\sigma_{\delta\theta}^{(0)}\lesssim\sigma_{\theta_{\mathrm{res}}}$, then the
EPIC telescopes can adjust their path extensions of eqs. 2.25 and 2.26 to have
the reference angle $\boldsymbol{\mathbf{\theta}}_{ba}^{\mathrm{ref}}$ within
about one fringe (or a few fringes) of the true separation
$\boldsymbol{\mathbf{\theta}}_{ba}$. In this case, the light-centroiding
sensitivity estimates of section 2.4 apply directly.

The majority of stars and quasars on which EPIC is applicable should have good
global astrometry in the Gaia catalog [22], which is essentially complete up
to the apparent magnitudes $m_{V}\lesssim 15$ of interest. Gaia DR4 and DR5
will provide time-averaged sky localization expected at the
$\mathcal{O}(20\,\mathrm{\mu as})$ level for these bright sources [91].
Relative astrometry for two sources at this precision will typically be better
than Phase I’s resolution, and not much worse than the optimal resolutions of
Phase II and III. This prior knowledge should thus be sufficient to
unambiguously perform extended-path intensity correlation, i.e. determine
$\boldsymbol{\mathbf{\theta}}_{ba}\cdot\boldsymbol{\mathbf{d}}$ from the set
of excess fractional intensity correlations $C$ in each spectral channel [1].

To understand how an EPIC measurement would work in practice, let us study the
time dependence of the correlator over the course of an observing night. We
choose variable path extensions $\ell_{1}^{\mathrm{opt}}(t)$ and
$\ell_{2}^{\mathrm{opt}}(t)$ according to eqs. 2.25 and 2.26 so that the
$\boldsymbol{\mathbf{\theta}}_{ba}^{\mathrm{ref}}(t)$ is consistent with the
prior astrometry on the source from Gaia. Due to the latter’s finite
precision, we will generally not satisfy eq. 2.1 _exactly_ , and instead have
a time-dependent difference:

$\displaystyle\boldsymbol{\mathbf{\theta}}_{ba}(t)\cdot\boldsymbol{\mathbf{d}}(t)-\left[\ell_{1}^{\mathrm{opt}}(t)+\ell_{2}^{\mathrm{opt}}(t)\right]$
$\displaystyle=\delta\boldsymbol{\mathbf{\theta}}_{ba}(t)\cdot\boldsymbol{\mathbf{d}}(t),$
(3.2)

with
$\delta\boldsymbol{\mathbf{\theta}}_{ba}(t)=\boldsymbol{\mathbf{\theta}}_{ba}(t)-\boldsymbol{\mathbf{\theta}}^{\mathrm{ref}}_{ba}(t)$
of characteristic size $\sigma_{\delta\theta}^{(0)}$. The differential
astrometric information in an EPIC observation enters through the fringes in
eq. 2.28, which are proportional to:

$\displaystyle\cos\left[k\delta\boldsymbol{\mathbf{\theta}}_{ba}(t)\cdot\boldsymbol{\mathbf{d}}(t)\right]$
$\displaystyle=\cos\left(k\left[\delta\theta_{ba}^{\delta}d_{z}\cos(\mathrm{dec})+\mathcal{A}\sin\left(\frac{2\pi
t}{\mathrm{day}}+\Phi\right)\right]\right),$ (3.3)

where $\mathcal{A}$ is defined as
$\mathcal{A}^{2}\equiv\left(d_{x}^{2}+d_{y}^{2}\right)\left[\left(\delta\theta_{ba}^{\alpha}\right)^{2}\cos^{2}(\mathrm{dec})+\left(\delta\theta_{ba}^{\delta}\right)^{2}\cos^{2}(\mathrm{ra})\right]$,
and $\Phi$ as
$\tan\Phi\equiv\left[-\delta\theta_{ba}^{\alpha}d_{y}\cos(\mathrm{dec})+\delta\theta_{ba}^{\delta}d_{x}\sin(\mathrm{dec})\right]/\left[{\delta\theta_{ba}^{\alpha}d_{x}\cos(\mathrm{dec})+\delta\theta_{ba}^{\delta}d_{y}\sin(\mathrm{dec})}\right]$.
The component(s) $d_{z}$ ($d_{x}$, $d_{y}$) is the baseline projection towards
the celestial North pole (lie in celestial equator plane), and “ra” and “dec”
refer to the right ascension and declination of the source pair, while
superscripts $\alpha$ and $\delta$ denote components along ra and dec.

Figure 10: The expected excess fractional intensity correlation between two
sources $a,b$ of equal brightness, as a function of time and wavelength. We
assume a mismatch of the reference position
$\boldsymbol{\mathbf{\theta}}_{ba}^{\mathrm{ref}}$ and the true position
$\boldsymbol{\mathbf{\theta}}^{ba}$ of
$\delta\theta^{\alpha}_{ba}=\delta\theta^{\delta}_{ba}=20\,\mathrm{\mu as}$
(left) and $100\,\mathrm{\mu as}$ (right), baseline vector components
$(d_{x},d_{y},d_{z})=(1\,\mathrm{km},0.5\,\mathrm{km},0.5\,\mathrm{km})$,
spectral resolution $\mathcal{R}=5{,}000$, and $\sigma_{t}=10\,\mathrm{ps}$.

As eq. 3.3 makes clear, the time variation in the argument of the correlator
is coming almost exclusively from the term with amplitude $\mathcal{A}$
(proportional to the magnitude of the baseline in the equatorial plane) and
sidereal frequency. The phase contribution from this change over an
observation time $t_{\mathrm{obs}}$ is approximately:

$\displaystyle\mathcal{O}(1)\times\left(\frac{\sqrt{d_{x}^{2}+d_{y}^{2}}}{1\,\mathrm{km}}\right)\left(\frac{500\,\mathrm{nm}}{\lambda}\right)\left(\frac{\sqrt{(\delta\theta^{\alpha})^{2}\cos^{2}\theta_{a}^{\delta}+(\delta\theta^{\delta})^{2}\sin^{2}\theta_{a}^{\delta}}}{20\,\mathrm{\mu
as}}\right)\left(\frac{t_{\mathrm{obs}}}{10\,\mathrm{hours}}\right).$ (3.4)

In other words, for baselines at non-vanishing latitudes and with reference
positions with Gaia uncertainties, one can expect order-unity variation in the
intensity correlation throughout an observing night, with smaller wavelengths
experiencing faster “fringe scanning”. We depict this effect in figure 10,
where over the course of 24 hours, the excess fractional intensity correlation
in each spectral channel would sweep from left to right. In practice, one
would chop up the observation period into shorter intervals, such that the
correlation is constant in each interval separately. The source separation
$\delta\boldsymbol{\mathbf{\theta}}_{ba}$ can be extracted from fitting to the
correlation in these time intervals across all spectral channels.

In some cases, the prior knowledge on the source separation may be
significantly worse than the angular resolution of the EPIC pair of
telescopes, but practically always better than their dynamic range:
$\sigma_{\theta_{\mathrm{res}}}\ll\sigma_{\delta\theta}^{(0)}<\sigma_{\Delta\theta}$.
Poorly known separations can occur when sources are within
$0.3\,\mathrm{arcsec}$ of each other (e.g. photometric or spectroscopic
binaries, double stars, or unresolved multiple images of a strongly lensed
quasar) so that they are typically not resolved in standard imaging
telescopes, either due to diffraction or seeing limitations. Because the map
$C\mapsto\overline{k}\boldsymbol{\mathbf{d}}\cdot\boldsymbol{\mathbf{\theta}}_{ba}$
is multivalued with roughly $\mathcal{R}$ fringes, a measurement on the
correlation $C$ in each spectral channel can correspond to
$\mathcal{O}(\sigma_{\delta\theta}^{(0)}/\sigma_{\theta_{\mathrm{res}}})$
physically viable solutions, no matter how high the signal-to-noise ratio.
This (discrete) degeneracy in parameters can, in principle be broken by
combining the information from all spectral channels, but this procedure can
be computationally demanding given the large data volume and the motion of
Earth and the source, both of which change the dot product angle
$\hat{\boldsymbol{\mathbf{d}}}\cdot\boldsymbol{\mathbf{\theta}}_{ba}$.

In a large _array_ of EPIC telescopes, one can perform a logarithmic search
for the separation vector $\boldsymbol{\mathbf{\theta}}_{ba}$. One could use a
pair of nearby EPIC telescopes (with small $d$ or serendipitously small
$\boldsymbol{\mathbf{d}}\cdot\hat{\boldsymbol{\mathbf{\theta}}}_{ba}$) whose
angular resolution $\sigma_{\theta_{\mathrm{res}}}^{(1)}$ is of order the
prior uncertainty $\sigma_{\delta\theta}^{(0)}$ on the source separation; this
first pair of EPIC telescopes would then improve the relative light-
centroiding precision to $\sigma_{\delta\theta}^{(1)}$, an improvement over
$\sigma_{\theta_{\mathrm{res}}}^{(1)}\sim\sigma_{\delta\theta}^{(0)}$ by a
factor roughly equal to the SNR (see eq. 2.50). A second pair of telescopes
with a larger projected baseline
$\boldsymbol{\mathbf{d}}\cdot\hat{\boldsymbol{\mathbf{\theta}}}_{ba}$ and
corresponding resolution
$\sigma_{\theta_{\mathrm{res}}}^{(2)}\sim\sigma_{\delta\theta}^{(1)}$ could
then improve the light-centroiding precision to $\sigma_{\delta\theta}^{(2)}$,
and so on. An EPIC array could thus “zoom in” on the source separation quite
efficiently, reaching its ultimate operating point near the main fringe with
only a logarithmically large number of steps, since the relative centroiding
precision improves as $\mathcal{O}(\mathrm{SNR}^{n})$ after $n$ iterations.
Fortuitously, this should be possible in Reuleaux triangle, circular, and
Y-shaped array configurations, which have centrally condensed response
functions, i.e. dense Fourier coverage at small spatial frequencies [87, 90].
This protocol would be robust but bring some overhead to the observation time
of any new pair of unresolved sources.

###  3.3 Photodetectors

The foremost requirement for (extended-path) intensity interferometry is a
very fast photon counter. The fringe contrast scales as $C\propto
1/\sigma_{t}$ and the overall SNR as $\propto 1/\sqrt{\sigma_{t}}$, where
$\sigma_{t}$ is the standard deviation for the differential photon arrival
time, which includes the timing resolution of both detectors, readout
electronics, and synchronization errors, cfr. eqs. 2.8. The timing precision
of photodetectors is conventionally quantified by their time jitter
$t_{\mathrm{jitter}}$, the full width at half maximum (FWHM) of the signal
curve as a function of time. For ease of comparison, we assume the signal
curve is Gaussian, and take the standard deviation for a single detector’s
timing resolution to be $t_{\mathrm{res}}=t_{\mathrm{jitter}}/(2\sqrt{2\ln
2})$. Assuming the readout electronics are not the bottleneck and both
telescopes share the same type of photodetector, then the _relative_ timing
precision from eq. 2.8 will be $\sigma_{t}\simeq
t_{\mathrm{jitter}}/(2\sqrt{\ln 2})\approx 0.72\,t_{\mathrm{jitter}}$.

In addition, we require that the photodetectors have a sufficiently small
reset time $t_{\mathrm{reset}}$, i.e. the “dead time” the detector needs to
revert to its original state after a single-photon absorption event. The
classes of photodetectors with the best timing jitters typically cannot
distinguish between single-photon or multiphoton events (in contrast to
photon-number-resolving devices). A reset time (much) smaller than the typical
time separation between photon events (in each spectral channel) is thus
necessary to have a high duty cycle and not wash out intensity fringe
correlations. For most sources under consideration and our assumed telescope
parameters (table 1), sub-microsecond reset times $t_{\mathrm{reset}}\lesssim
1\,\mu\mathrm{s}$ should be sufficient, see figure 11 for example.

Thirdly, we desire a high quantum efficiency $\eta_{p}$ (cfr. eq. 2.43) to
maximize the SNR (eq. 2.49) and light-centroiding precision (eq. 2.51). The
fact that each photodetector pixel is only responsible for reading out a
narrow bandwidth is advantageous in this regard, as each pixel can be tuned to
maximize the photodetection efficiency in that particular spectral channel.

Finally, photodetectors in general will have spurious events at rate known as
the dark count rate (DCR), which may arise intrinsically (e.g. from electronic
noise) or environmentally (from cosmic rays, blackbody radiation, stray light,
etc.). Some environmental backgrounds can be partially shielded by an
appropriate spectral filter on the pixels, as each pixel of the photodetector
only needs to read out a narrow band. However, the tolerances on the DCR are
somewhat loose, as the relatively large photon flux from the sources of
interest constitutes our main source of statistical noise: in the sea of
incoming photons, we are searching for those very rare photon pairs that
simultaneously (after correcting for propagation differences) impinge on both
telescopes. The DCR is not the limiting factor when below the sources’
combined photon flux. If larger, it would reduce the intensity correlation
fringe contrast, as the DCR would give false positives that are uncorrelated
between the two telescopes.

The above four requirements point towards to two possible technologies for
single-photon detectors, namely superconducting nanowire single photon
detectors (SNSPDs) and single photon avalanche diodes (SPADs), also known as
Geiger mode avalanche photodiodes (GmAPD). The present state of the art for
these two classes of devices is summarized below.

Figure 11: Rate of recorded photons per spectral channel incident on the
aperture of one telescope. We use the same parameters as in figure 6 for the
source pairs and Phase II detectors (also in table 1). The upper shaded region
corresponds to photon rates in excess of $10^{6}\,\mathrm{s}^{-1}$, where
significant sensitivity loss would occur (due to missed photon counts) for a
photodetector reset time of $t_{\mathrm{reset}}=1\,\mathrm{\mu s}$. Likewise,
the lower shaded region corresponds to source photon rates below the dark
count rate (DCR) of the photodetector, where loss of fringe contrast and thus
sensitivity would occur because dark counts from electronic noise are
uncorrelated. If $N_{\mathrm{subap}}$ subapertures are used, then photon rates
are reduced by a factor of $N_{\mathrm{subap}}^{-1}$, weakening the reset time
constraint but tightening the DCR requirement.

Given the demonstrated technology and the real-word applications of both
SNSPDs and SPADs outside pristine laboratory conditions, we are optimistic
that arrays of photodetectors tailored to the needs of multichannel intensity
interferometry are feasible. In figure 11, we plot the photon rate in a single
spectral channel of an EPIC Phase II telescope, for the same benchmark source
pairs as in figure 6, showing that these rates fall between achievable DCRs
and inverse reset times $t_{\mathrm{reset}}^{-1}$. Ultrafast photon detection
over a large array of pixels (each optimized at different wavelengths) has not
been a priority of most current research efforts, but it is clear from our
summaries below that the requisite technology exists, and that there is no
obstacle preventing its further development. Our proposal offers unique
prospects for using this cutting-edge technology for a novel use case in
astronomy, and we hope the exciting scientific applications (section 4) will
further intensify research and development in this area.

#### Superconducting Nanowire Single Photon Detectors

Single photon detection with superconducting nanowires, first developed about
two decades ago, is a technique with extraordinary promise for quantum optics
and many other applications, owing to picosecond-level timing jitter, sub-
millihertz DCR, near-unity quantum efficiency, sub-nanosecond recovery times
[92, 93], and ability to operate from the infrared (IR) to optical range [94]
and even in the ultraviolet (UV) [95]. The physical principle upon which
SNSPDs operate is simple: after absorption of a single photon by the
superconducting nanowire, a large number of Cooper pairs are broken, inducing
a local hotspot in the normal, non-superconducting phase. A bias current is
applied so that this hotspot subsequently expands due to Joule heating,
further increasing the resistance, generating a voltage pulse that can be
amplified and registered with room-temperature electronics. The energy of the
hotspot is made to dissipate quickly thereafter to return the nanowires back
to their superconducting state.

The system timing jitter for single SNSPDs has steadily improved over the last
decade, from $18\,\mathrm{ps}$ ($15\,\mathrm{ps}$ intrinsic) [96] to as low as
$2.6\,\mathrm{ps}$ for $400\,\mathrm{nm}$ and $532\,\mathrm{nm}$ visible light
with a niobium nitride (NbN) nanowire at $0.9\,\mathrm{K}$ operating
temperatures [94], with possible further improvements. Recently, SNSPDs have
made further advances, from the UV to the IR. A UV SNSPD with an active area
of $56\,\mathrm{\mu m}$ in diameter was demonstrated to have a timing jitter
down to $62\,\mathrm{ps}$, a system efficiency of up to 84% for narrowband
designs, a DCR of $0.25\,\mathrm{counts/hr}$, and detector dead time less than
$250\,\mathrm{ns}$ [95]. On the other side of the spectrum, a free-space
coupled NbN SNSPD was shown to have a $14\,\mathrm{ps}$ time jitter, a DCR
below $0.1\,\mathrm{Hz}$, a system efficiency of up to $84\%$, with an active
area of $22\times 15$ $\mu$m2 at $1550\,\mathrm{nm}$ wavelengths [97].
Optimization of the system’s design and fabrication can push photodetection
efficiencies as high as 98% [98], and with tungsten silicide (WSi) nanowires,
the energy threshold can be lowered to perform single-photon detection at
$10\,\mathrm{\mu m}$ wavelengths.

Multiplexing SNSPDs into arrays with many pixels—while simultaneously
retaining their exquisite performance—presents new challenges. To date, the
largest realized SNSPD array consists of $32\times 32$ pixels with a row-
column multiplexing architecture [99], optimized at $1550\,\mathrm{nm}$ with
an area of $1.6\times 1.6\,\mathrm{mm^{2}}$ ($0.96\times
0.96\,\mathrm{mm}^{2}$ active area) and operated at $0.73\,\mathrm{K}$, which
exhibited a timing jitter of at least $250\,\mathrm{ps}$. Due to the small
filling factor and low pixel efficiency, the combined system efficiency was
only 8%, but there are clear paths to improve these factors [99], and other
thermal-coupling array designs may be scaled into the megapixel range [100].
The array cannot distinguish between photons that impinge on the detector with
a relative arrival time less than the timing jitter. Saturation of the
photodetector array restricts the maximum apparent brightness of sources, as
we illustrate in figure 11. We should note that lab measurements using this
array were done over second-long timescales, so scaling up to hours of
observation is yet to be demonstrated. Finally, we note that a _single_
nanowire alone can serve _multiple_ pixels using time-of-flight information:
ref. [101] reports a $50\,\mathrm{ps}$ timing jitter at $405\,\mathrm{nm}$ and
MHz photon counts, with one wire resolving 590 effective pixels. Large SNSPD
arrays with active areas up to $3.1\times 3.1\,\mathrm{mm^{2}}$ have been
considered for use in exoplanet transit spectroscopy [102] (e.g. on the
Origins space telescope [103]), for which exceptional photometric stability is
required in the IR range of $2.8$–$20\,\mathrm{\mu m}$.

#### Single Photon Avalanche Photodiodes

A SPAD is a diode operated well above its reverse-bias breakdown voltage.
Because of the very large internal electric field, a single photoelectron is
accelerated to energies sufficiently high to ionize other electrons in its
path, thus creating a self-sustaining avalanche, which is registered as a
“click” (hence the Geiger mode monicker). The detector’s dead time is set by
the timescale for quenching the avalanche current and restoring it back to the
operating bias.

Already in 2008, single SPADs $20\,\mathrm{\mu m}$ in diameter could reach
efficiencies up to 42% with a timing jitter of $39\,\mathrm{ps}$, kHz-level
DCR, and reset times of about $600\,\mathrm{ns}$ [104]; ten years later, these
performance numbers were improved with a $10$–$80\,\mathrm{\mu m}$ diameter
bipolar-CMOS-DMOS SPAD to $t_{\mathrm{jitter}}\approx 28\,\mathrm{ps}$,
$\eta=70\%$, $t_{\mathrm{reset}}=50\,\mathrm{ns}$ at $820\,\mathrm{nm}$ and
operating temperature of $-20^{\circ}\,\mathrm{C}$ [105]. The envelope
continues to be pushed, with reports of $8.7\,\mathrm{ps}$ timing jitters up
to MHz count rates [106], and further improvements on the horizon.

SPADs have been assembled into arrays as large as $256\times 256$, with
demonstrated timing jitters of $300\,\mathrm{ps}$, reset times of
$100\,\mathrm{ns}$, $70\%$ efficiency, and DCR of $10\,\mathrm{Hz}$ [107].
These arrays can be manufactured to support Gbit/s data rates (relevant for
telecommunications), which is on the order of the photon fluxes relevant even
for Phase 3 dishes of EPIC. [107]. The Europa Lander astrobiology mission is
to be equipped (as part of its LIDAR system) with a $2048\times 32$ array of
Si SPADs with a DCR of $4.5\,\mathrm{kHz}$, peak detection efficiency of 70%
at $532\,\mathrm{nm}$ [108], and timing jitter target of $250\,\mathrm{ps}$
[109]; a smaller $1024\times 32$ version for the Europa Lander Sensor Array
brought the latter figure down to $33\,\mathrm{ps}$ [110].

###  3.4 Dispersive element

As discussed in section 3.3, despite major leaps in timing resolution of
modern single-photon counters, the relative timing uncertainty $\sigma_{t}$ is
still much larger than the period of electromagnetic waves in the optical
band. The fringe contrast in intensity interferometry increases with
decreasing spectral width $\sigma_{k}$ (eq. 2.11). For a (point-like) source
with a broad spectrum, we thus have $C\sim 1/c\sigma_{k}\sigma_{t}$, but the
flux is reduced as $I\propto\sigma_{k}$. For a _single_ spectral channel, this
implies that the SNR is independent of $\sigma_{k}$ (eq. 2.48). However, by
combining _multiple_ spectral channels of width $\sigma_{k}$ across an
$e$-fold around wavenumber $\overline{k}$, the signal-to-noise ratio can be
enhanced as
$\mathrm{SNR}\propto(\overline{k}/\sigma_{k})^{1/2}=\mathcal{R}^{1/2}$, as
derived in eq. 2.49. The most natural way to spectrally split the incoming
light is via a dispersive prism or a diffraction grating, as in modern
spectrographs. In the following, we perform a simplified analysis of this
spectroscopic procedure and note some associated challenges unique to
intensity interferometry.

The main (and perhaps surprising) result of this section is that the intensity
correlation for a multichannel intensity interferometer can receive an
additional suppression for large-aperture telescopes and excellent spectral
and timing resolution. Without mitigating solutions discussed below, the
excess fractional intensity correlation in any one spectral channel
parametrically scales as:

$\displaystyle
C\sim\min\left\\{1,\frac{1}{c\sigma_{k}\sigma_{t}},\frac{\theta_{\mathrm{diff}}}{\theta_{\mathrm{seeing}}}\right\\},$
(3.5)

which is also depicted in figure 12. Comparing to e.g. eq. 2.11, the
fractional correlation is limited by the ratio of the diffraction-limited
angle $\theta_{\mathrm{diff}}\equiv{2\pi}/{\overline{k}D}$ of a single
telescope to the seeing angle $\theta_{\mathrm{seeing}}$ (eq. 2.63). This
correlation ceiling would be the limiting factor for the large-aperture
telescopes with high spectral resolution and exquisite timing resolution in
all three EPIC Phases of table 1. We conclude this section with possible
solutions that can recover the sensitivity assumed throughout section 2, i.e.
remove the ${\theta_{\mathrm{diff}}}/{\theta_{\mathrm{seeing}}}$ ceiling in
eq. 3.5.

In order to illustrate this potential suppression as lucidly as possible, we
study the following simplified setup: a plane wave incident on a grating of
size $Nd_{\mathrm{g}}$, with $d_{\mathrm{g}}$ the periodicity of the grating
and $N$ the number of (illuminated) slits. We concern ourselves only with the
first diffraction peak focused onto a photodetector array, of which a single
pixel records the wavenumbers
$k\in\left[\bar{k}-\sigma_{k}/2,\bar{k}+\sigma_{k}/2\right]$, with
$\sigma_{k}\equiv\bar{k}/\mathcal{R}$. In appendix C, we carry out the full
calculation within scalar diffraction theory in great detail, from the point
of incidence on the primary mirror to the photodetector array.

The electric field at wavenumber $k$ far away from a diffraction grating is
given by the standard result [111]:

$E_{k}(\theta_{\text{F}},t)\propto
A_{k}\frac{\sin\left(k\theta_{\text{F}}{Nd_{\mathrm{g}}}/{2}\right)}{\sin\left(k\theta_{\text{F}}{d_{\mathrm{g}}}/{2}\right)}e^{-ikt},$
(3.6)

with $\theta_{F}\equiv x_{\text{F}}/f_{c}$, where $x_{\text{F}}$ is the
physical linear coordinate on the photodetector, $f_{\text{c}}$ the focal
length of the camera which focuses the collimated light from the grating onto
the photodetector, and $A_{k}$ the field amplitude at wavenumber $k$. The
first diffraction peak at wavenumber $\overline{k}$ occurs at
$\theta_{\mathrm{F}}\simeq\overline{\theta}_{F}\equiv 2\pi/(\overline{k}d)$
(in the small-angle approximation). We are free to redefine
$\theta_{\mathrm{F}}\to\overline{\theta}_{\mathrm{F}}+\theta_{\mathrm{F}}$ so
that $\theta_{\mathrm{F}}=0$ corresponds to the center of the pixel
corresponding to the spectral channel centered at $\overline{k}$. This pixel
subtends a small angular width equal to $\overline{\theta}_{F}/\mathcal{R}$.
With this redefinition of $\theta_{\mathrm{F}}$ and expanding $\sin x\simeq x$
at $x\ll 1$, the electric field has the form:

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

Writing the intensity correlation as
$C=\frac{\left<I_{1}(0)I_{2}(t)\right>}{\left<I_{1}\right>\left<I_{2}\right>}-1\equiv\frac{\mathcal{N}}{\mathcal{D}}$,
we find for the numerator $\mathcal{N}$ and denominator $\mathcal{D}$:

$\displaystyle\mathcal{N}$
$\displaystyle\propto\int\frac{\text{d}t}{\sigma_{t}}\iint\text{d}\theta_{\text{F}}^{(1)}\text{d}\theta_{\text{F}}^{(2)}\iint\text{d}k\,\text{d}k^{\prime}E_{k}(\theta_{\text{F}}^{(1)},0)E_{k^{\prime}}^{*}(\theta_{\text{F}}^{(1)},0)E_{k}^{*}(\theta_{\text{F}}^{(2)},t)E_{k^{\prime}}(\theta_{\text{F}}^{(2)},t),$
(3.8) $\displaystyle\mathcal{D}$
$\displaystyle\propto\left|\int\text{d}\theta_{\text{F}}\int\text{d}k\,\left|E_{k}(\theta_{\text{F}},0)\right|^{2}\right|^{2},$
(3.9)

where the superscript $(p)$ labels the telescope $p=1,2$ location of the
pixel. For simplicity, we incorporate the relative timing uncertainty via a
uniform distribution of width $\sigma_{t}$. Both expressions contain four
factors of the electric field in eq. 3.7. The numerator involves products of
electric fields with different wavenumbers $k$ and $k^{\prime}$ at each
telescope. In contrast, the denominator is simply the product of the average
intensities at each telescope, involving products of electric fields with the
_same_ wavenumber at each telescope. A heuristic understanding of the
suppression effect is thus that large pixels—larger than the diffraction
limit—contain several spatially non-overlapping electromagnetic modes with
_independent_ temporal fluctuations, so that $C=\mathcal{N}/\mathcal{D}$ does
_not_ attain the limit $C\to 1$ as $\sigma_{t}\to 0$.

We can gain some (analytic) intuition by taking the size of the pixels to be
large (formally infinite) while simultaneously endowing $A_{k}$ with (uniform)
support only over a wavenumber range of width $\sigma_{k}$.222In practice, the
spectral width $\sigma_{k}$ of the pixel is linearly proportional to its
angular size, so this limit is not physical—we only provide it for
illustrative purposes. In this limit, which turns out to be a reasonable
approximation, we find (see appendix C.4):

$C=\frac{\mathcal{N}}{\mathcal{D}}\simeq\frac{2}{\sigma_{k}^{2}}\int_{0}^{\sigma_{k}}\text{d}\sigma\,(\sigma_{k}-\sigma)e^{-\frac{1}{2}\sigma^{2}\sigma_{t}^{2}}\left[\frac{\sin\sigma/\sigma_{d}}{\sigma/\sigma_{d}}\right]^{2}.$
(3.10)

This integral consists of a convolution of two effective “filters”: one
Gaussian filter with a bandwidth $\sigma_{t}^{-1}$, and one sinc-squared
filter with bandwidth $\sigma_{d}\equiv\bar{k}/(\pi N)$. In the limit of poor
timing resolution, $(c\sigma_{t})^{-1}\ll\sigma_{d}$, the latter filter is
nearly unity over the integration range, and we recover the standard result
$C\sim 1/{c\sigma_{k}\sigma_{t}}$ (from e.g. eq. 2.11). However, in the
opposite limit $(c\sigma_{t})^{-1}\gg\sigma_{d}$ of good timing resolution, it
is the Gaussian filter that is unity, yielding
$C\sim\pi\sigma_{d}/\sigma_{k}\ll 1/c\sigma_{k}\sigma_{t}$.

Figure 12:  Intensity correlation of eq. 3.10 for a single pixel whose central
wavelength is $\overline{\lambda}=500\,\mathrm{nm}$, as a function of the
effective linear size of the photocollection element, and assuming a seeing
angle of $\theta_{\mathrm{seeing}}=1\,\mathrm{arcsec}$. We plot this
correlation for the timing and spectral resolutions $\sigma_{t}$ and
$\mathcal{R}$ assumed for Phases I, II, and III of EPIC. The horizontal axis
can be read either as the diameter of the primary mirror (vertical black
dashed lines) for a setup with only one photodetector array, or as the linear
subaperture size in the spectroscopic direction (see “Potential Solutions”
below). The spectroscopic suppression of
$C\sim\theta_{\mathrm{diff}}/\theta_{\text{seeing}}$ (gray dashed line) is the
limiting factor for $D\gtrsim{2\pi
c\sigma_{t}}/{\mathcal{R}\theta_{\mathrm{seeing}}}$. The ideal subaperture
size is roughly this threshold value: $1\,\mathrm{m}$, $0.2\,\mathrm{m}$, and
$0.1\,\mathrm{m}$ for EPIC Phases I, II, and III, corresponding to a number of
subapertures of $N_{\mathrm{subap}}\sim 4,50,100$.