# Coriolis Force Compensation and Laser Beam Delivery for 100-Meter Baseline
Atom Interferometry

Jonah Glick  Department of Physics and Astronomy and Center for Fundamental
Physics, Northwestern University, Evanston, Illinois 60208, USA  Zilin Chen
Department of Physics and Astronomy and Center for Fundamental Physics,
Northwestern University, Evanston, Illinois 60208, USA  Tejas Deshpande
Department of Physics and Astronomy and Center for Fundamental Physics,
Northwestern University, Evanston, Illinois 60208, USA  Yiping Wang
Department of Physics and Astronomy and Center for Fundamental Physics,
Northwestern University, Evanston, Illinois 60208, USA  Tim Kovachy
Department of Physics and Astronomy and Center for Fundamental Physics,
Northwestern University, Evanston, Illinois 60208, USA
<EMAIL_ADDRESS>
###### Abstract

The Coriolis force is a significant source of systematic phase errors and
dephasing in atom interferometry and is often compensated by counter-rotating
the interferometry laser beam against Earth’s rotation. We present a novel
method for performing Coriolis force compensation for long-baseline atom
interferometry which mitigates atom-beam misalignment due to beam rotation, an
effect which is magnified by the long lever arm of the baseline length. The
method involves adjustment of the angle of the interferometer beam prior to a
magnifying telescope, enabling the beam to pivot around a tunable position
along the interferometer baseline. By tuning the initial atom kinematics, and
adjusting the angle with which the interferometer beam pivots about this
point, we can ensure that the atoms align with the center of the beam during
the atom optics laser pulses. This approach will be used in the MAGIS-100 atom
interferometer and could also be applied to other long-baseline atom
interferometers. An additional challenge associated with long baseline
interferometry is that since long-baseline atom interferometers are often
located outside of typical laboratory environments, facilities constraints may
require lasers to be housed in a climate-controlled room a significant
distance away from the main experiment. Nonlinear effects in optical fibers
restrict the use of fiber-based transport of the high-power interferometry
beam from the laser room to the experiment. We present the design of and
prototype data from a laser transport system for MAGIS-100 that maintains
robustness against alignment drifts despite the absence of a long fiber.

††preprint: AIP/123-QED

##  I Introduction

Atom interferometry is a powerful tool for testing fundamental physics Fray
_et al._ (2004); Bonnin  _et al._ (2013); Bouchendira  _et al._ (2011);
Tarallo  _et al._ (2014); Schlippert  _et al._ (2014); Kuhn  _et al._ (2014);
Rosi  _et al._ (2014); Barrett  _et al._ (2015); Biedermann  _et al._ (2015);
Zhou  _et al._ (2015); Barrett  _et al._ (2016); Williams  _et al._ (2016);
Rosi  _et al._ (2017); Asenbaum  _et al._ (2017); Overstreet  _et al._ (2018);
Parker  _et al._ (2018); Asenbaum  _et al._ (2020a); Morel  _et al._ (2020);
Overstreet  _et al._ (2022); Hamilton  _et al._ (2015). Atom intererometers
with 100-meter-scale baselines and larger provide an opportunity to detect
gravitational waves in a frequency range between 0.01 Hz and 3 Hz Dimopoulos
_et al._ (2008); Graham  _et al._ (2013, 2016a); Chaibi  _et al._ (2016);
Hogan  _et al._ (2011); El-Neaj  _et al._ (2020); Abe  _et al._ (2021), search
for ultra-light, wave-like dark matterArvanitaki  _et al._ (2018); Graham  _et
al._ (2016b); Banerjee  _et al._ (2022), and test quantum mechanics over
macroscopic spatio-temporal scalesKovachy  _et al._ (2015); Abe  _et al._
(2021). Excitement about the potential scientific utility of long-baseline
atom interferometry has spurred the ongoing construction of multiple long-
baseline atom interferometers, including AIONBadurina  _et al._ (2020),
MIGACanuel  _et al._ (2018), ZAIGA Zhan  _et al._ (2019), VLBAIHartwig  _et
al._ (2015), and MAGIS-100Abe  _et al._ (2021). The sensitivity of an atom
interferometer generally increases with an increasing baseline length, but
larger baselines introduce new technical challenges. In this paper, we address
two key challenges associated with scaling up the baseline of an atom
interferometer: 1. Traditional methods of compensating for systematic effects
associated with the rotation of the Earth break down for longer baselines and
2. Delivering the interferometer beam from the climate-controlled laser room
to the main experiment with alignment that is robust against drifts can be
more difficult in areas such as deep shafts because of facilities constraints.
For instance, this is a major challenge faced by MAGIS-100.

Coriolis forces caused by the rotation of the Earth induce velocity dependent
phase shifts in an atom interferometer Dickerson  _et al._ (2013); Lan  _et
al._ (2012). Long baselines can enable long temporal spacing $T$ between beam
splitter and mirror pulses, as well as large momentum transfer (LMT), but the
scale of the Coriolis induced phase shifts increases with increasing $T$ and
LMT number $n$. These phase shifts can wash out the interferometer contrast
when the signal is extracted by averaging over the atomic ensemble. One
solution to mitigate this effect is that in the limit in which the atom cloud
can be approximated as expanding from a point source, the Coriolis phase
shifts result in spatial fringes across the cloud which can be measured with
spatially resolved detection Dickerson  _et al._ (2013). Even in this limit,
however, there are two challenges posed by these velocity dependent phase
shifts: 1. the Coriolis force can cause shot-to-shot fluctuations in the
average initial kinematics of the cloud (e.g., from jitter in the atom cooling
and launching processes) to manifest as fluctuating shot-to-shot overall phase
shifts, and 2. As $T$ and $n$ increase, even in the point source limit,
fringes could be of such a high spatial frequency that they are difficult to
spatially resolve in detection. It is therefore important to adopt strategies
to compensate for Coriolis forces.

A method to compensate for Coriolis forces in atom interferometers has been
successfully demonstrated and consists of rotating the interferometer beam
against the rotation of the Earth with a single piezo-actuated mirror
Dickerson  _et al._ (2013); Lan  _et al._ (2012). This scheme breaks down for
longer interferometer baselines because the long lever arm between the piezo-
actuated mirror and the atoms can cause the center of the interferometer beam
to transversely shift away from the atom cloud as the beam is rotated. In
section II, we present a new method to allow for Coriolis force compensation
in longer baseline atom interferometers. This method keeps the atoms in the
center of the interferometer beam while the beam is pulsed on and performing
beam splitter or mirror atom optics operations. We also consider challenges
associated with using this method to do Coriolis force compensation for km-
scale baseline interferometers.

Additionally, while atom interferometers have to date often been located
inside single rooms in a laboratory, 100-meter-scale and larger atom
interferometers will tend to require facilities of a much larger scale such as
deep shafts. Building these facilities can be cost prohibitive, but adapting
existing facilities is more feasible. Existing facilities of this type were
typically not built with atom interferometry in mind and can therefore pose
constraints that need to be overcome. In MAGIS-100, for example, the atom
interferometry lasers will be located in a dedicated laser room to ensure
stable environmental temperatures and to meet laser safety requirements, but
space constraints require this laser room to be located approximately 10 m
from the shaft that contains the atom interferometer. For distances of this
scale, stimulated Brillouin scattering Kobyakov, Sauer, and Chowdhury (2010)
significantly complicates the prospects for using a single-mode optical fiber
to transport the high-power interferometry laser beam from the laser room to
the shaft Abe  _et al._ (2021).

A key advantage of fiber-based transport is that the position and pointing of
the post-transport beam is determined solely by the fiber, which can be
anchored to a structurally stable floor. On the other hand, free-space beam
delivery is more susceptible to alignment drifts, especially with a relatively
long $\approx 10\;$m lever arm. In Section III, we discuss the design of a
laser transport system for MAGIS-100 which uses a combination of stable
mechanical mounts, a relay imaging system, and a short fiber to provide
delivery of the interferometer beam to the shaft that maintains the
aforementioned benefits of delivery via a long fiber but avoids the
limitations of stimulated Brillouin scattering. In Section III.2 we present
data from a prototype test of this system. The stability of the alignment of
the final delivery optics before the 100 m baseline is also critical, but this
is aided by the presence of a structurally stable floor and wall by the shaft,
and by the fact that the telescope which magnifies the size of the beam also
demagnifies beam angles, reducing the impact of angular drifts from pre-
telescope optics.

##  II  Pivot Point Method for Rotation Compensation

###  II.1  Outline of the method

It has been demonstrated in a 10 m baseline atom interferometer that counter-
rotating an interferometer beam against the rotation of the Earth suppresses
interferometer phase shifts and contrast loss from the coupling of Coriolis
forces to the initial kinematics of the atom cloud Dickerson  _et al._ (2013);
Lan  _et al._ (2012); Asenbaum  _et al._ (2020b); Sugarbaker  _et al._ (2013).
Traditionally, this counter-rotation scheme has been done with a single piezo-
actuated tip-tilt mirror at one end of the baseline, but this approach breaks
down for larger interferomter baselines. The location of this mirror sets the
position about which the interferometer beam pivots. For larger baseline
interferometers, the increased lever arm can cause the center of the
interferometer beam to move away from the center of the atom cloud as it is
rotated, which in the extreme case causes the beam to miss the atom cloud
entirely, and can otherwise cause pulse inefficiency and phase errors due to
inhomogeneous intensities and laser phases across the cloud. If the position
of the atoms is a distance $L$ from the pivot point of the interferometer beam
then, to the atoms, the interferometer beam appears translated by an amount
$\approx L\theta$ where $\theta$ is the angle of the interferometer beam. For
a 10 m baseline atom interferomter at the latitude of Batavia, Illinois (the
location of the MAGIS-100 experiment at Fermi National Accelerator Laboratory)
the scale of this translation
$L\theta=(L)(\Omega_{\oplus}\cos[\theta_{\text{lat}}])(T)\approx 0.7\text{
mm}$, which is small relative to the size of a cm-scale interferometry laser
beam, and the fact that the atoms see a slightly changed intensity profile
during the course of an experiment cycle has a small impact on the measurement
outcome. We have defined $\Omega_{\oplus}$ to be the rotation rate of the
earth, and $\theta_{\text{lat}}$ to be the latitude of Batavia. Here we have
taken $L=10\text{ m}$, $\Omega_{\oplus}=7.29\times 10^{-5}\text{ rad/s}$,
$\theta_{\text{lat}}=41.85^{\circ}\times\frac{\pi}{180^{\circ}}\text{ rad}$,
$T=1.3\text{ s}$, and the radial waist of the interferometer beam to be
$1\;\text{cm}$ Abe  _et al._ (2021). The increased lever arm and longer
interrogation times associated with 100 m baseline interferometry (e.g.,
$L=100\text{ m}$, $T=4\text{ s}$) increases the scale of this translation to
$\approx 2.2\;\text{cm}$, which is larger than the radial beam waist, so an
alternate scheme for rotation compensation is required in order to keep the
atom cloud centered on the interferomter beam in the presence of rotation
compensation. One solution is to make the size of the beam larger–however,
with laser power limited by practical constraints, increasing the beam size
lowers the Rabi frequency, which has two detrimental effects: 1. Smaller Rabi
frequncies leave atom optics pulse efficiencies more susceptible to the
Doppler detuning spread of the atom cloud, and 2. The achievable Rabi
frequency can limit the number of pulses that can be driven in a given time,
especially for single-photon atom optics on relatively weak clock transitions,
which can limit the number of LMT enhancement pulses that can be performed in
the interferometer.

Here, we argue that if one can tune the position about which the inteferometer
beam pivots along the axis of the interferometer, and one has control over the
initial position and launch angle of the atom cloud, rotation compensation can
be achieved in such a way that the atoms stay centered with the center of the
interferometer beam during the times that the laser is pulsed on and
interacting with the atoms even when the size of the beam is small compared to
the translational range spanned by the beam during an experiment cycle. The
capability of tuning the initial cloud position and launch angle for
individual atom sources is built into MAGIS-100Abe  _et al._ (2021).

Figure 1:  The optical configuration associated with the pivot point method:
By actuating the angle of the interferometer beam a distance $d_{a}$ from the
first telescope lens, the beam pivots about a point that is a distance $d_{p}$
from the second telescope lens. A subsequent mirror is angled so that the
interferometer beam is retro-reflected. The telescope length $L$ is taken to
be the sum of the focal lengths which compose the telescope, $L=f_{1}+f_{2}$.
In this conceptual illustration, the scale of $L$ is on the order of the scale
of $d_{p}$, but in the MAGIS-100 beam delivery system, $d_{p}$ will in general
be one order of magnitude larger than $L$.

The optical configuration associated with this pivot point scheme consists of
having piezo-actuated mirrors on each end of the baseline and tuning the path
length of the interferometer beam before a telescope to set the position of
the pivot point. In MAGIS-100, a telescope will magnify the beam waist to
approximately $1\;\text{cm}$ before the beam interacts with the atoms. We
define $d_{a}$ to be the distance between the first telescope lens and the
point about which the pointing of the interferometer beam is actuated (the
‘actuation point’). The position of the pivot point can be adjusted by
adjusting $d_{a}$, as discussed in section II.3. As indicated in Figure 1, the
‘pivot point’ is defined as the position after the telescope for which angular
adjustments about the ‘actuation point’ manifest as purely angular adjustments
in the beam, with no associated translations. A mirror further along the
beam’s propagation path rotates with the rotating beam so that the retro-
reflected beam is colinear with the incident beam.

In Section II.2, we outline the implemenation of the pivot point scheme for
the case of performing interferometery over a $100\text{ m}$ baseline with a
single atom source, as in the case of one of the dark matter search operating
modes of MAGIS-100. In this operating mode, co-located, simultaneous atom
interferometers using different isotopes of strontium will span the length of
the baselineAbe  _et al._ (2021). In Section II.6, we extend the scheme to the
case of performing atom gradiometry over $100\;\text{m}$ baselines, as in the
gravitational wave detection operation mode of MAGIS-100. In Section II.7, we
explore the prospects for, and challenges of, scaling this scheme to km-scale
baselines.

Figure 2:  The relationship between the unit vectors which define the lab
frame Cartesian coordinate system $(\hat{x},\hat{y},\hat{z})$ and the rotation
vector of the earth $\vec{\Omega}_{\oplus}$ as seen in (a) the space frame,
and (b) the lab frame. In the limit that the angles of the interferometer beam
$\delta\\!\theta$ required to counter-rotate against the rotation of the earth
are small, $\delta\\!\theta\ll 1$, the required rotations are primarily around
the $y$ axis.

We argue that to first order in small angular deviations, it suffices to
consider rotations about a single axis orthogonal to the vertical direction.
Figure 2 shows the relationship between the unit vectors which define the lab
frame Cartesian coordinate system $(\hat{x},\hat{y},\hat{z})$ and the rotation
vector of the Earth $\vec{\Omega}_{\oplus}$. Since the axis about which the
Earth rotates is not orthogonal to the interferometer axis, $\hat{z}$,
perfectly rotating the interferometer beam against the rotation of the Earth
would in general require rotations about the axis
$\vec{\Omega}_{\oplus}=\Omega_{y}\hat{y}+\Omega_{z}\hat{z}$ corresponding to
the rotation vector of the Earth. For small enough interrogation times $T$,
the required rotations of the interferometer beam are primarily around the $y$
axis. Here we choose the coordinate system so that the interferometer axis is
along $\hat{z}$, the $y$ axis points toward the north pole, and the $x$ axis
points eastward. The components of Earth’s rotation vector in these
coordinates have magnitudes
$\Omega_{y}=|\vec{\Omega}_{\oplus}|\cos[\theta_{\text{lat}}]$ and
$\Omega_{z}=|\vec{\Omega}_{\oplus}|\sin[\theta_{\text{lat}}]$. Consider a
rotation matrix $\hat{R}_{\oplus}[\delta\\!\theta]$ which performs rotations
by a small angle $\delta\\!\theta$ around the vector $\vec{\Omega}_{\oplus}$.
To study the scale of the required rotations of the interferometer beam, we
take the beam to be initially pointing along the $z$ axis, $\hat{k}=\hat{z}$,
and consider the effect of a small Coriolis force compensating rotation on
$\hat{k}$, $\hat{k}^{\prime}=\hat{R}_{\oplus}[\delta\\!\theta]\hat{z}$. To
first order in $\delta\\!\theta$, the required rotation of the interferometer
beam is solely about the $y$ axis, in the $x-z$ plane:
$\hat{k}^{\prime}=\hat{R}_{y}\big{[}\cos[\theta_{\text{lat}}]\delta\\!\theta\big{]}\hat{z}+\mathcal{O}[\delta\\!\theta^{2}]$,
where $\hat{R}_{y}[\theta]$ is a matrix which rotates a vector an angle
$\theta$ about the $y$ axis.

For a single Mach-Zehnder interferometer with interrogation time $T=4\text{
s}$, which is indicated in Figure 3 and which represents an upper bound on the
value of $\delta\\!\theta$ which would be required in MAGIS-100,
$\delta\\!\theta=|\vec{\Omega}_{\oplus}|T\approx 0.29\text{ mrad}$ with
$|\vec{\Omega}_{\oplus}|\approx 7.29\times
10^{-5}\frac{\text{rad}}{\text{s}}$. At this maximum angular range, the angle
that is made by the $z$ axis and the projection of $\hat{k}^{\prime}$ onto the
$x-z$ plane is
$\cos[\theta_{\text{lat}}]\delta\\!\theta+\mathcal{O}[\delta\\!\theta^{2}]\approx
0.22\text{ mrad}$, while the angle that is made by the $z$ axis and the
projection of $\hat{k}^{\prime}$ onto the $z-y$ plane is
$\frac{1}{2}\sin[\theta_{\text{lat}}]\cos[\theta_{\text{lat}}]\delta\\!\theta^{2}+\mathcal{O}[\delta\\!\theta^{3}]\approx
21\text{ nrad}$. For simplicity of illustration in figures, we will ignore the
small projection of $\hat{k}^{\prime}$ onto the z-y plane, though the rotation
compensation system will have the capability to implement the corresponding
small corrections to the angle of $\hat{k}^{\prime}$ as needed.

In considering the overlap between the interferometry laser beam and the
atoms, it is also important to consider deflections of the atom trajectories
due to Coriolis forces. The predominant deflections in the trajectories are in
the $x-z$ plane because the Coriolis force couples the motion of an atom along
the $x$ axis with the large ‘upward’ component of its launch velocity,
$v_{z}$, and does not couple the $y$ component of the motion to $v_{z}$ since
the axis of rotation is in the $y-z$ plane. We estimate residual deflections
along the $y$ axis for the scheme indicated in Figure 3 to be $\approx
4.5\;\mu\text{m}$. To simplify the illustration of the pivot point method, we
do not consider here dynamics of the atom trajectory in the $y$ direction and
focus on the $x-z$ plane. We note also that the centrifugal force produces a
constant acceleration in the $y-z$ plane which can be thought of as
contributing a small correction to a pure gravitational acceleration:
$\vec{g}_{\text{eff}}=(g-R_{e}\Omega_{y}^{2})\hat{z}+(R_{e}\Omega_{y}\Omega_{z})\hat{y}$,
where $R_{e}$ is the radius of the Earth (taking the Earth to be a perfect
sphere)Morin (2008). In practice, the interferometer beam will be aligned to
the direction in which the atoms freely fall, which is along
$\vec{g}_{\text{eff}}$. Taking this effect into account, the $\approx
4.5\;\mu\text{m}$ deflection cited above can be understood as referring to a
deflection along a $y^{\prime}$ axis, where the primed coordinates refer to a
coordinate system which has been slightly rotated around the $x$ axis so that
the $z^{\prime}$ axis points along $\vec{g}_{\text{eff}}$.

###  II.2 Coriolis Force Compensation for Long Baseline Interferometry with a
Single Atom Source

In Figure 3, a Mach-Zehnder pulse sequence with a $T=\;4\text{ s}$ spacing
between beam splitter and mirror pulses and LMT number $n=1000$ is performed,
wherein an atom cloud is launched at an angle of $\approx 0.14\text{ mrad}$
from the $z$ axis with a total velocity of $\approx$ 35.86 m/s. The launch
angle is chosen so that the two arms of the interferometer have transverse
position $x=0$ at the time of the mirror pulse, and so that the position of
the atom cloud is the same for both the first and final beam splitter times.
The radius of the earth is taken to be $R_{e}$ = 6378 km. At time t = 0, an
LMT beamsplitter pulse is performed by an interferometer beam angled by
$\Omega_{y}T\approx 0.22\text{ mrad}$ about the origin of the lab frame
coordinate system which imparts a momentum kick on the upper interferometer
arm of $1000\hbar k$ (for simplicitly, we assume that the momentum kicks
happen instantaneously), where $k$ is the wave number of the 679 nm light that
is resonant with the ${}^{3}P_{0}\leftrightarrow\\!\\!\\!\ ^{3}S_{1}$
resonance of strontium (for the dual-isotope operating mode of MAGIS-100, atom
optics will be performed via two-photon Bragg transitions on the 679 nm
resonance of strontium Abe  _et al._ (2021)). The atoms then propagate in a
superposition of different momentum states subject to Coriolis and centrifugal
forces before the mirror pulse at time $t=T$ imparts a $1000\hbar k$ momentum
kick downward on the upper interferometer arm and a kick with equal magnitude
and opposite direction on the lower arm. The atoms then freely propagate for a
time $T$ before spatially recombining, at which point a final beamsplitter
pulse is performed by the beam angled symmetrically about the $z$ axis with
respect to its angle at time $t=0$. The choice of pivot point and initial atom
cloud kinematics allows for the beam to stay centered on the atom cloud for
each interaction time while the interferometer beam is rotated to suppress
Coriolis dependent phase shifts and contrast loss. In this example, the pivot
point is located at the initial position of the atoms at $t=0$.

Figure 3:  A diagram demonstrating the basic pivot point idea for a Mach-
Zehnder sequence. (a) The angle of the interferometer beam at the time of the
beamsplitter pulse $t=0$, the mirror pulse $t=T$, and the final beamsplitter
pulse $t=2T$, along with the paths traced by the classical trajectories of the
two arms of the interferometer at each of those three times. (b) The total
classical paths traversed by the upper and lower interferometer arms along
with each of the three angles of the interferometer beam. For the dual-isotope
operating mode of MAGIS-100, the momentum kicks will be achieved via two-
photon Bragg transitions on the 679 nm resonance of strontium Abe  _et al._
(2021). In this case, the blue line denotes the classical path traces by the
part of the atom wavefunction in the $|0\hbar k\rangle$ state which has not
been kicked by any photons, and the red line denotes path traces by the atom
in the $|1000\hbar k\rangle$ state that has been kicked by 1000 photons. The
radial beam waist of the beam is 1 cm, and the rays are separated by a
distance of 1/6 cm.

###  II.3 Tuning the Location of the Pivot Point

The location of the pivot point can be tuned by adjusting the distance between
the actuation point and the first telescope lens, $d_{a}$ (see Fig. 1). Here
we perform ABCD matrix ray tracing calculations in the paraxial and thin lens
limits in order to provide an approximate analytic scaling of the pivot point
location as a function of $d_{a}$. We consider physical optics effects,
including also the effects of the exact lens geometries, with numerical
calculations using the Zemax software package in section II.5. Here we
consider an optical system consisting of two telescope lenses with focal
lengths $f_{1}$ and $f_{2}$ separated by a distance $f_{1}+f_{2}$. A ray
pivots a distance $d_{a}$ away from the first telescope lens, and we define
$d_{p}$ to be the distance from the second telescope lens to the pivot point.
The relationship between $d_{p}$ and $d_{a}$ can be expressed as

$d_{p}=(f_{2}+M^{2}f_{1})-M^{2}d_{a}$ (1)

where $M=f_{2}/f_{1}$ is the magnification of the telescope and the other
parameters are indicated in Fig. 1. For MAGIS-100, two ultrahigh quality
spherical lenses (Optimax) Abe  _et al._ (2021) with focal lengths
$f_{1}=150\text{ mm}$ and $f_{2}=4500\text{ mm}$ will be used to magnify a
beam focused to a $\approx 300\;\mu\text{m}$ radial waist at the position of
the tip tilt mirror to a waist of $\approx 1\text{ cm}$. A vacuum-compatible,
piezo-actuated tip-tilt mirror mount has been procured from Mad City Labs (a
custom vacuum-compatible version of the Nano-MTA2) which will enable tilting
the interferometer beam before this telescope. This mount has a total angular
range of $10\text{ mrad}$, and for tilts about $\hat{y}$–which will be made to
be approximately orthogonal to the nominal plane of incidence–the total range
with which the beam can be angled is $20\text{ mrad}$ because the change in
the angle of an incident beam upon reflection from the actuatable tip-tilt
mirror is twice the angle of incidence. The $M=30$ telescope will magnify the
size of the interferometer beam by a factor of $M$ and de-magnify the angles
produced by this tilted mirror by that same factor, so that the largest
angular tilt allowed by this system in one direction is $\approx 10\text{
mrad}/M\approx 0.33\text{ mrad}$. This angular range is large enough to
perform the $T=4\text{ s}$ sequence shown in Figure 3, whose maximum angular
range is $\approx 0.22\text{ mrad}$ in one direction. The tip-tilt retro-
reflection mirrorAbe  _et al._ (2021) positioned $\approx 100\text{m}$ below
this second telescope lens will be angled to keep the reflected beam aligned
with the downgoing beam.

###  II.4  Tip-Tilt System Mechanical Design

As shown in Figure 4, our design of this section of the tip-tilt system
involves the piezo-actuated stage mounted at a $45^{\circ}$ angle on a steel
post. Out of vacuum stepper motors will adjust the distance between the tip
tilt mirror and the first telescope lens to set the pivot point, and the
piezos in the Mad City Labs mount will rotate the mirror during an experiment
cycle for Coriolis force compensation. We will have the capability to
translate the beam with out-of-vacuum stepper motors, which could be useful
for fine tuning the alignment of this beam.

Figure 4:  CAD model of the in-vacuum tip-tilt mirror and first telescope
lens.

###  II.5 Effect of Spherical Lens Aberrations

In this section, we determine the scale of the aberrations associated with the
interplay of the pivot point system and the spherical nature of the telescope
lenses through physical optics simulations using the Zemax software package,
which capture non-paraxial effects along with effects from the finite extent
of the curved face of the telescope lenses. We study the aberrations induced
by the spherical telescope lenses on the intensity and phase profiles of the
interferometer beam under $0$ and $10\text{ mrad}$ beam tilts at five
different propagation distances between $0$ and $200\text{ m}$ after the
second telescope lens. We find that the scale of the aberrations induced on
the interferometer beam by the spherical nature of the telescope lenses are
comparable to the expected phase deviations from surface imperfections in the
optics themselves.

Spherical lenses are cheaper to manufacture but induce some anharmonicity into
the phase profile of a transmitted Gaussian beam. The dominant mechanism by
which aberrations in the interferometer beam are expected to limit the
sensitivity of the MAGIS-100 interferometer is the coupling of aberrations to
laser pointing jitter and to shot-to-shot fluctuations in initial atom cloud
kinematics Abe  _et al._ (2021).

To investigate aberrations associated with the interplay of the tip/tilt
system and the telescope, we start with a Gaussian beam profile at a focus
with a radial waist $300\;\mu\text{m}$ incident on the tip/tilt mirror. The
first lens has a plano-convex shape and is oriented with its curved face
pointing away from the center of the telescope a distance $50\text{mm}$ from
the tip-tilt mirror. The radius of curvature of this lens is
$R_{1}=68.295\text{mm}$, and its center thickness is $t_{1}=3\text{ mm}$. The
second telescope lens is also plano-convex and also has its curved face
pointing away from the center of the telescope. This lens has radius of
curvature $R_{2}=2048.85\text{mm}$ and a center thickness of $t_{2}=7.5\text{
mm}$. The distance between the curved faces of the lenses is equal to the sum
of their focal lengths $L=f_{1}+f_{2}=4650\text{ mm}$, where
$f=\frac{R}{n-1}$. Both lenses are made of a material whose index of
refraction is approximately $n=1.455$ (fused silica). The parameter $d$ in
Figures 5 and 6 denotes the distance from the curved face of the second
telescope lens to the plane in which the intensity and phase profiles are
evaluated. Here we plot only the cross section of the phase and intensity
profiles along the $x$ axis for different distances $d$ along the $z$ axis,
which for rotations about the $y$ axis is where the dominant sources of
aberration are under rotations of the beam.

Figure 5:  (a) Cross sections of interferometer beam (a) phase profiles and
(b) intensity profiles after propagating through the spherical telescope
lenses with no initial tilt (solid line) and with a $10\text{ mrad}$ tilt
$5\text{ cm}$ before the first telescope lens (dotted line) for a few
different propagation distances $d$ after the second telescope lens. Here the
linear component of the tilted beam’s phase profile, computed via geometric
optics, is ignored. $\Delta\phi_{L}[x,y]$ is defined to be the phase profile
of the interferometer beam, where $x$ and $y$ are the dimensions transverse to
the propagation axis of the beam, and $I[x,y]$ is defined to be the intensity
profile of the beam Figure 6:  (a) The difference in the phase profiles and
(b) the relative difference in intensity profiles at different propagation
distances $d$ after the second telescope lens between the untilted
interterferometer beam and a beam tilted by $10\text{ mrad}$ a distance of
$5\text{ cm}$ before the first telescope lens. Here the difference is taken
after the tilted interferometer beam is shifted in $x$ at each propagation
distance $d$ by the expected position of the beam along $x$ as determined by
geometric optics.

For ease of visualization, the phase profile plots ignore a phase factor
varying linearly in $x$ that corresponds to the angle of the beam as
calculated purely by ray tracing. We see no remarkable aberrations in the
intensity profile of the beam, and the aberrations to the phase profile, as
seen in Figure 6, are on length scales on the order of the beam size. Over a
$\pm 9\text{ mm}$ range centered on the center of the beam, the phase
aberrations have a maximum magnitude of $\approx 12\text{ mrad}$, which is
comparable to the expected phase deviations from surface imperfections in the
optics themselves. The size of the atom cloud is $\approx 3\text{ mm}$ or
less, so beyond this range, most atoms will not sample the inhomogeneity in
the laser phase profile.

###  II.6  Coriolis Compensation for Atom Gradiometry

An atom gradiometer is an apparatus in which multiple atom interferometers are
spaced over a long baseline. MAGIS-100 is designed to be able to operate in
this mode for mid-band gravitational wave detection and certain dark matter
searches Abe  _et al._ (2021). The influence of Coriolis forces can be
suppressed in a gradiometer configuration by performing multiloop sequences
Dubetsky and Kasevich (2006); Abe  _et al._ (2021), but this approach also
suppresses the response of the interferometer to low frequency signals. The
pivot point method outlined in Section II.1 could be used in a gradiometer
mode where the signals of interest are of a lower frequency. Performing long
baseline rotation compensation with the pivot point method for atom
gradiometry relies on the ability to adjust the initial positions and
velocities of the different atom clouds individually. In Figure 7, three atom
interferometers are separated by 45 m and span roughly the length of the 100 m
baseline. The sequence presented in Figure 7 is a Mach-Zehnder interferometer
with a $T=1\text{ s}$ interrogation time and $160\hbar k$ momentum transfer at
each atom-laser interaction time. The atom clouds from bottom to top are
launched with respective angles of approximately $35.09\;\mu\text{rad}$,
$-0.23\text{ mrad}$, and $-0.49\text{ mrad}$ relative to the $z$ axis, each
with a total initial velocity of $\approx 9.26\text{ m/s}$. The interferometer
beam is initially angled by $\Omega_{y}T\approx 54.3\;\mu\text{rad}$, and the
atom clouds are prepared with a lateral offset from the interferometer
baseline of $\approx 0,2.44\text{ mm},\text{and }4.87\text{ mm}$, read from
bottom to top.

Figure 7:  Illustration of rotation compensation with the pivot point method
in gradiometer mode. Three atom interferoemters are initially located at
respective vertical positions of $z_{0}=0\text{ m},45\text{ m},90\text{ m}$.
The spacing between beam splitter and mirror pulses if $T=1$ s, and 160$\hbar
k$ LMT is used. Note that the horizontal axis for each interferometer has a
different range. In this example, the pivot point is chosen to be at the
location of the bottom atom source as in Figure 3, but the pivot point could
instead be located at the middle atom source to reduce the size of the
required initial position offset of the topmost atom cloud.

###  II.7 Prospects For, and Challenges of, Scaling Up to Longer Baselines

The pivot point method can be extended to future larger baseline
interferometers, but alternate telescope parameters are required in order to
place the pivot point near the end of the longer baseline and ensure the
ability to tune it over a range about that point. Only two-lens telescopes are
considered here, though telescopes with three or more lenses may also be
feasible. Figure 8 presents a few possible lens combinations, referencing
Equation (1). If the range over which $d_{a}$ is adjusted is held fixed as the
baseline increases, a larger telescope magnification is required in order to
scan the pivot point over a larger range. In general, as one increases the
baseline of the interferometer, a longer two-lens telescope is required. One
challenge associated with performing Coriolis force compensation with the
pivot point method over longer baselines is that the required deflections of
the interferometer beam are larger, which requires vacuum pipes with larger
internal diameters. The diameter of the second telescope lens also increases
with increasing baseline length. For a Mach-Zehnder interferometer sequence
wherein the classical trajectories of the atoms span km-scale baselines, the
required rotations about the pivot point are roughly $\Omega_{y}TL\approx
0.7\text{ m}$ for $L=1\text{ km},\;T=13\text{ s}$, and $\Omega_{y}\approx
54.3\;\mu\text{rad}/\text{s}$. In order for the interferometer beam to not
clip on the edge of optics or on the internal diameter of vacuum pipes, the
bottom telescope lens and the internal diameter of the vacuum pipes near the
second telescope lens would need to have diameters on the meter scale, which
would have significant associated costs.

Figure 8:  Possible $d_{p}[d_{a}]$ for different choices of $f_{1}$ and
$f_{2}$, referencing Equation (1). The dependence of $d_{p}$ on $d_{a}$ for
three different lens combinations are considered here in order to illustrate
some possible lens combinations for interfereomters with baselines up to the
km scale. The length of the telescope $L$ is defined to be the sum of the
focal lengths of the two lenses which compose the telescope, $L=f_{1}+f_{2}$.
The solid line corresponds to the MAGIS-100 telescope parameters, and the
dotted lines are potential lens combinations for longer baseline
interferometers.

##  III Interferometer Laser Beam Delivery System

###  III.1  Beam Delivery System Mechanical Design

The laser transport system transports the interferometer beam from where it is
generated in the laser room over to the shaft where the interferometry is
being done. It uses a combination of stable mechanical mounts, a relay imaging
system, and a short fiber to provide both passive pointing stability and a
stable multi-Watt interferometer beam power.

Building a dedicated laser room serves two critical functions: temperature
control and laser safety. Having a dedicated temperature-controlled room
suppresses associated drifts and reduces the frequency of various laser
feedback loops falling out of lock. Also, consolidating the laser system in a
central, interlocked room is required for laser safety purposes. Constructing
a laser room within a facility not purpose-built for atom interferometry
presents a challenge: It can be difficult to position the laser room in close
proximity to the main experiment without resorting to costly and time-
consuming structural modifications to the existing facility in order to
accommodate the laser room’s spatial footprint. Consequently, this constraint
may necessitate situating the laser room at a considerable distance from the
main experiment. MAGIS-100 is being constructed in the MINOS Service Building
at Fermilab, which was originally built as an access building for underground
experiments. Facilities constraints require the interferometry laser beam to
be transported to the shaft over $\approx 10\text{ m}$ horizontally, and also
for the beam to be transported at a height of $\approx 3\text{ m}$ above the
floor of the building.

A limitation of fiber-based laser transport is that for multi-Watt laser
powers and $\approx 10\text{m}$ long fibers, losses from stimulated Brillouin
scattering become a significant concernKobyakov, Sauer, and Chowdhury (2010).
It is productive to maximize the power of the beam that is delivered to the
atoms because larger beam powers result in larger Rabi frequencies, which are
advantageous for the reasons outlined in Section II.1. We therefore opt to
transport the beam in free space. Free space transport, however, poses its own
challenges since it is more susceptible to alignment drifts in the delivery
optics.

To address these challenges, we include a $\approx 1\text{ m}$ long, single-
mode fiber (short enough that we do not observe stimulated Brillouin
scattering) after the free space laser transport to remove the influence of
pointing fluctuations from all preceding optics and free-standing support
structures. We will then anchor all subsequent optics to a structurally stable
floor and wall by the shaft. With this short fiber, pointing fluctuations in
the beam manifest as fluctuating fiber coupling efficiencies. After the short
fiber, we will pick off a small amount of light to measure the post-fiber
power of the beam and feed that signal back to the laser room, where an
acousto-optic modulator will adjust and stabilize this power. The lock point
must be set slightly below the maximum achievable laser power in order to
correct for dips in the post-fiber power of the beam caused by drifts in the
laser transport. Improved stability of the laser transport system suppresses
the magnitude of these power drops, allowing for a higher lock point, and an
increase in the power delivered to the atoms. This combination of a short
fiber and a power stabilizing feedback system enables the beam to have both
stable pointing and power.

Figure 9:  The idea behind relay imaging: Lenses in the beam path reduce the
susceptibility of the fiber coupling efficiency to drift in optics. Without
the relay imaging lenses, the long distance between fibers causes small
angular drifts in the beam to manifest as large translations of the beam
incident on the second fiber, and therefore large coupling efficiency falloff.
The relay imaging lenses suppress this effect.

The delivery optics will drift on hour timescales in response to temperature
changes in the area, and fluctuate on faster timescales in response to seismic
and anthropogenic vibrations in the environment, along with air currents. To
suppress the influence of pointing fluctuations on the fiber coupling
efficiency, we opted to add two long focal length lenses in a ‘4f’ relay-
imaging configuration to better passively isolate the coupling efficiency of
the interferometer beam against pointing fluctuations of the steering optics,
effectively reducing the lever arm associated with the beam propagation (see
Fig. 9). Moreover, the interferometer beam propagates in rough vacuum
($\approx 1\text{ Torr}$) to suppress the influence of air currents on the
pointing of the beam. We tested the efficacy of using a relay imaging system
to stabilize the fiber coupling efficiency in a full-scale prototype of the
laser transport system (see Sec. III.2 for further details).

Figure 10:  Optical schematic of the full beam delivery system. See main text
for details. Figure 11:  CAD model of the beam delivery system whose optical
configuration is illustrated in Figure 10.

The ‘shaft-side optics’ (see Figures 10 and 11) consist of the short fiber
described above, which provides initial spatial filtering of the beam Abe  _et
al._ (2021), a photodetector for power monitoring (used for power
stabilization feedback as described above), optics for a fiber-phase-noise-
cancellation setup, and a short telescope. One fixed mirror on the shaft-side
breadboard will direct the interferometer beam to a short telescope, which
shapes the beam to match the mode of the short fiber. Two subsequent piezo-
actuated mirror mounts adjust the angle and lateral position of the
interferometer beam into the fiber. These mounts can be adjusted every
$\approx 40\text{ hrs}$ to compensate for long timescale drifts in the laser
transport system alignment. The polarizing beamsplitter and mirror just after
the fiber back-reflect a small portion of the interferometer beam as part of a
fiber-phase-noise-cancellation scheme DeRose  _et al._ (2023); Ma  _et al._
(1994); Abe  _et al._ (2021). A subsequent Faraday isolator keeps preceding
optics protected from back-reflected light, and then a short telescope shapes
the interferometer beam so that it is focused to a radial waist of $\approx
300\;\mu\text{m}$ at the location of the tip-tilt mirror, $\approx 4.5\text{
m}$ away. Two subsequent mirrors mounted to motorized translation stages
provide fine adjustment of the lateral alignment of the interferometer beam
into the telescope region. Adjusting the lateral position of the
interferometer beam may also be useful to keep the beam centered on atoms for
extended LMT sequences when the effective mirror and beamsplitter pulses take
a non-negligible amount of time.

The ‘telescope region’ (see Figure 10) exists in UHV ($10^{-9}\text{ Torr}$)
and consists of $\approx 4.5\text{m}$ of initial free propagation (1 m of
horizontal propagation and 3.5 m of vertical), which further cleans the
spatial mode of the beamAbe  _et al._ (2021). The beam then reflects off the
tip-tilt mirror, whose mechanical configuration is discussed earlier in
Section II.4, and travels through the $M=30$ telescope discussed in section
II.5 which expands the interferometer beam to a waist of $\approx 1\text{
cm}$. The piezo-actuated mirror stage at the bottom of the shaft Abe  _et al._
(2021) retro-reflects the interferometer beam back up the shaft. The telescope
de-magnifies angles by a factor equal to its magnification $M$, reducing the
impact of angular fluctuations of pre-telescope optics.

###  III.2 Full Scale Prototype of Laser Transport System

We constructed a prototype of the laser transport system to demonstrate the
feasibility of coupling the interferometer beam into a short fiber following
$\approx 20\text{ m}$ of free propagation and to monitor the passive stability
of the fiber coupling efficiency over time scales of many hours in the
presence of environmental perturbations. This test was performed in a
horizontal configuration (see Fig. 12) on the rooftop of the Fermilab
Accelerator Science and Technology accelerator in the New Muon Lab on the
Fermilab campus and included the right-angle turns that will be necessary to
elevate the beam off the floor in the final configuration. The relay imaging
system was composed of two f = 5 m lenses in a ‘4f’ configuration respectively
mounted to two vacuum chambers which each housed a fixed right angle mirror.
This proof of concept was without any of the vertical support structures
planned for the full laser transport system, since those structures were still
being designed. As shown in Figure 13, over a continuous 40-hour period, we
successfully maintained a $>80\%$ fiber coupling efficiency without the need
for any alignment adjustments.

Figure 12: An image and optical schematic of the prototype test of the laser
transport system. See main text for details.

The prototype consisted of a ‘launching’ breadboard which consisted of a
$\approx 5\text{ mW}$ laser (CivilLaser) which was coupled into a fiber to
clean the spatial mode of the beam, and directed to a right angle mirror and 5
m focal length lens (EKSMA 110-0289E) $\approx 5\;\text{m}$ away. The beam
then propagated $\approx 10$ m, before reflecting off a second right angle
mirror paired with a second 5 m focal length lens. Subsequently, the beam
propagated over a $\approx 5\text{ m}$ distance to the ‘receiving’ breadboard.
On this board, three mirrors set the alignment of the beam into a short
telescope which mode-matched the beam into a short fiber. A 50:50 beamsplitter
and photodiode combination just before the fiber monitored the power of the
beam before the fiber, and a photodiode after the fiber measured the post-
fiber power. To extract a coupling efficiency, we took the ratio of the real-
time power before and after the fiber so that the measured coupling efficiency
was independent of power fluctuations in the laser.

We obtained best stability results when using Thorlabs Polaris mirror mounts
(POLARIS-K1S4) with the pitch and yaw lead screw axes locked with locking
nuts. To suppress the higher frequency jitter induced by air currents, the
beam was enclosed by PVC pipe in the long regions of free-space propagation,
and cardboard boxes with two holes cut in them served as local enclosures for
the right angle mirrors. Both the launching and receiving breadboards were
enclosed with hardboard enclosure material from Thorlabs. While air current
shielding did not noticeably affect the long term stability, it slightly
suppressed root mean square (rms) fluctuations from $0.5\%$ to $0.4\%$
integrated over a 20 mHz to 4 kHz band. As shown in Figure 13, we measured a
$>80\text{\%}$ fiber coupling efficiency over a $40\text{ hr}$ period despite
this $\approx 20\text{ m}$ path length. To compensate for drifts on these
large timescales, automated tweaks of fiber coupling with piezo-controlled
mirror mounts, indicated in Figure 10, will be used.

Figure 13:  Trace of the fiber coupling efficiency as a function of time. The
raw data was sampled at a rate of 1 kHz, and the data as presented here is
averaged over 10 second bins.

In the near future, we plan to test the stability in the final vertical
configuration to evaluate the impacts of the vertical support structures.

##  IV Conclusion

In this paper, we introduced a method for Coriolis force compensation in long-
baseline atom interferometers, provided specific implementation details for
MAGIS-100, and presented the design of the beam delivery system for MAGIS-100.
In the future, we will demonstrate the pivot point method experimentally in
MAGIS-100 and further evaluate the feasibility of using this method for future
km-baseline interferometers.

###### Acknowledgements.

We thank Jason Hogan and James Santucci for valuable discussions and technical
contributions. This project is funded in part by the Gordon and Betty Moore
Foundation Grant GBMF7945. Some of the work presented in this document
leveraged the resources of the Fermi National Accelerator Laboratory
(Fermilab), a U.S. Department of Energy, Office of Science, HEP User Facility.
Fermilab is managed by Fermi Research Alliance, LLC (FRA), acting under
Contract No. DE-AC02-07CH11359. This work is supported in part by the U.S.
Department of Energy, Office of Science, QuantiSED Intitiative. We also
acknowledge support from the David and Lucile Packard Foundation (Fellowship
for Science and Engineering).

## Author Declarations

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Jonah Glick: Conceptualization (equal), Methodology (equal), Investigation
(lead), Validation (equal), Formal Analysis (lead), Writing – original draft
(lead). Writing – review & editing (equal). Zilin Chen: Methodology (equal),
Investigation (supporting), Validation (equal), Writing – review & editing
(supporting). Tejas Deshpande: Conceptualization (equal), Methodology (equal),
Investigation (supporting), Validation (equal), Supervision (equal), Writing –
review & editing (supporting). Yiping Wang: Conceptualization (equal),
Methodology (equal), Investigation (supporting), Validation (equal), Writing –
review & editing (supporting). Tim Kovachy: Conceptualization (equal),
Methodology (equal), Validation (equal), Supervision (equal), Writing – review
& editing (equal).

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Conflict of Interest Statement

The authors have no conflicts to disclose.

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11institutetext: University of Virginia, Charlottesville, Virginia, USA
11email<EMAIL_ADDRESS>

# MaNi: Maximizing Mutual Information for Nuclei Cross-Domain Unsupervised
Segmentation

Yash Sharma 11 Sana Syed Co-Corresponding Author11 Donald E.
Brown11footnotemark: 1 11

###### Abstract

In this work, we propose a mutual information (MI) based unsupervised domain
adaptation (UDA) method for the cross-domain nuclei segmentation. Nuclei vary
substantially in structure and appearances across different cancer types,
leading to a drop in performance of deep learning models when trained on one
cancer type and tested on another. This domain shift becomes even more
critical as accurate segmentation and quantification of nuclei is an essential
histopathology task for the diagnosis/ prognosis of patients and annotating
nuclei at the pixel level for new cancer types demands extensive effort by
medical experts. To address this problem, we maximize the MI between labeled
source cancer type data and unlabeled target cancer type data for transferring
nuclei segmentation knowledge across domains. We use the Jensen-Shanon
divergence bound, requiring only one negative pair per positive pair for MI
maximization. We evaluate our set-up for multiple modeling frameworks and on
different datasets comprising of over 20 cancer-type domain shifts and
demonstrate competitive performance. All the recently proposed approaches
consist of multiple components for improving the domain adaptation, whereas
our proposed module is light and can be easily incorporated into other methods
(Implementation: https://github.com/YashSharma/MaNi).

###### Keywords:

Unsupervised Domain Adaptation Contrastive Learning Histology Instance
Segmentation Semantic Segmentation

##  1 Introduction

Nuclei are the fundamental organizational unit of life. Accurate assessment of
these nuclei in histopathology slides provides critical morphological
information necessary for the quantitative analysis of multiple diseases. In
the last decade, Convolutional Neural Network-based approaches have emerged as
a promising solution in medical imaging tasks, and nuclei segmentation tasks
[7, 21]. However, they require a large amount of annotated data for any
reliable training. Moreover, even with a large dataset, models are prone to be
poor at generalizing learned knowledge from one cancer type to another. Pixel-
level image annotations by a medical expert are required to expand to a new
cancer type, making it difficult, expensive, and a time-consuming task.
Therefore, there is a need for methods that can transfer learned information
from existing cancer datasets to other domains without additional annotations.
We tackle this problem, characterized as Unsupervised domain adaptation (UDA),
to close the gap between the annotated source domain and unlabeled target
domain by learning domain-invariant and task-relevant features.

It has been observed that trivially training a model on the source domain and
evaluating on a target domain leads to sub-par performance owing to domain
shift [8]. Moreover, the domain shift problem is widespread in medical image
datasets due to different scanners, scanning protocols, and tissue types,
among others. Hence, multiple strategies comprising of adversarial learning,
pseudo-labeling, consistency regularization, or data-augmentation have been
proposed to tackle this limitation [8]. Among these, self-training has emerged
as a competitive approach. Various de-noising strategies such as confidence
thresholding, uncertainty estimation, or distance penalty between domains have
been proposed. These strategies have led to gains, but they all rely on
unlabeled data points closer to labeled data points for pseudo-label training
since those are the points with high confidence and attempts to push the
target distribution closer to the source. However, forcing the target-domain
distribution towards the source-domain distribution can destroy the latent
structural patterns of the target domain, leading to a drop in performance. We
hypothesize that instead of using these approaches, using an information-
theoretic distance can lead to better alignment between the domains.

Contrastive learning (CL) has seen wide success in representation learning
with applications ranging from unsupervised pre-training to multi-modal
alignment [22]. The basic idea of CL is to push together the latent
distribution of similar samples and push away the latent distribution of
dissimilar samples. Wang et al. [25] adopted a supervised, pixel-wise
contrastive learning algorithm and treated pixels belonging to the same class
as positive pairs and pixels from dissimilar classes as negative pairs,
observing significant gains in semantic segmentation performance. We take
inspiration from their work and expand the idea of contrasting similar class
pixels against dissimilar class pixels for UDA.

In summary, our paper makes the following contributions: 1) We propose a
simple Jensen-Shannon Divergence-based contrastive loss for UDA in nuclei
semantic and instance segmentation tasks. The proposed loss maximizes the
mutual information between the representations of ground truth nuclei pixels
from the labeled source dataset and the pseudo-labeled nuclei pixels in the
target data. 2) We demonstrate our approach using different architectures and
for over 20 cancer-type domain shifts establishing that the inclusion of the
MI loss leads to competitive gain over recently proposed methods.

##  2 Related Works

###  2.1 Unsupervised Domain Adaptation

UDA is a well-studied problem in literature where two of the widely adopted
techniques are adversarial learning and self-training. In Adversarial
learning, researchers attempt to align source data representation with target
data representation via discriminator training. Hoffman et al. [11] proposed a
cycle-consistent adversarial domain adaptation (DA) method for enforcing
cycle-consistency between domains. [17] used a category-level adversarial
network for enforcing semantic consistency between each class. Tsai et al.
[23] developed a patch alignment method for DA by adversarially pushing the
feature representations of clustered patches together. Vu et al. [24] used an
entropy-based adversarial training approach for aligning weighted self-
information distributions of different domains. Yang et al. [26] showed
improvement in domain alignment by iteratively defensing against pointwise
adversarial perturbations for domains.

In Self-training, pseudo-labels are iteratively generated on target datasets
using the model trained on labeled source datasets and used for retraining.
Recent papers have incorporated different de-noising strategies for improving
the accuracy of pseudo-labels. Zhang et al. [29] used category centroids from
the source domain for pseudo-labeling the target data and used the distance to
centroids for training. Zhang et al. [28] denoised pseudo-label by online
correcting them according to the relative feature distance to the prototypes.
Zou et al. [30] used class-wise normalized confidence scores for generating
pseudo-labels with balanced class distribution for self-training. Zou et al.
[31] proposed a confidence regularization for smoothing the prediction via
regularizer loss minimization.

We highlight the relevant works extending the above approaches for UDA in
nuclei segmentation. Yang et al. [27] used adversarial domain discriminator
and cyclic adaptation with pseudo labels for UDA. Further, they utilized weak
labels to improve nuclei instance segmentation and classification. Haq et al.
[9] used adversarial learning loss and reconstruction loss on output space for
UDA for cell segmentation. Li et al. [15] extended [9] and applied a self-
ensembling method with a student-teacher framework for imposing consistency
loss along with reconstruction and adversarial training. In another branch of
works, [16] performed UDA from microscopy to histopathology images by
synthesizing target-type images using CycleGAN followed by an inpainting
module before applying adversarial adaptation. However, this approach requires
pixel-level translation for synthesizing target-like images before a
segmentation module and uses instance-level information for the mutual
information-based feature alignment, constraining its adoption to only
instance segmentation. Therefore, we limit our work and first test it
extensively for tissue/ cancer type domain shifts and leave microscopy to
histopathology adaptation for future works. Further, our proposed MI-based
feature alignment strategy can be used for both semantic and instance
segmentation problems.

###  2.2 Contrastive Learning and Mutual Information

Contrastive learning (CL) methods are being widely adopted for classification
and segmentation tasks of different modalities. For semi-supervised
segmentation, Alonso et al. [1] used positive-only CL for enforcing the
segmentation network to yield similar pixel-level feature representation for
same-class samples between labeled and unlabeled datasets. Hu et al. [13] for
MRI and CT image segmentation pretraining used global contrastive loss on
unlabeled images and supervised local contrastive loss on limited labeled
images. For volumetric medical segmentation, Chaitanya et al. [2] leveraged
similarity across corresponding slices in different volumes for defining
positive and negative pairs for contrastive loss. They used infoNCE bound for
CL, requiring a large number of negative samples for training. For tackling
this dependency, Peng et al. [20] maximized mutual information (MI) on
categorical distribution by projecting the continuous feature embedding to
clustering space. They used encoder representation for global regularization
and, for local regularization, maximized MI between neighboring feature
vectors at multiple intermediate levels.

Our work maximizes the MI between similar classes using labeled source images
and pseudo-labeled target images. Concurrent to our work, Chaitanya et al. [3]
proposed an end-to-end semi-segmentation framework by defining a local pixel-
level contrastive loss between pseudo-labels of unlabeled sets and limited
labeled sets. They randomly sample pixels from each image to address the
computational limitations of running CL for all the pixels. In contrast, we
use average pooling at the class level for considering all the pixels of
labeled and unlabeled images. Further, we use JSD bound instead of InfoNCE for
MI estimation and focus on domain adaptation tasks. Shrivastava et al. [22] in
their work for visual representation learning from textual annotation,
demonstrated that the JSD-based bound enables the MI maximization at a smaller
batch size with just one negative sample. Also, [10] demonstrated in their
extended analysis that JSD is insensitive to the number of negative samples,
while infoNCE shows a decline as the number of negative samples decreases,
motivating us to choose JSD bound for MI estimation.

##  3 Methods

###  3.1 Problem Set-Up

In our work, we tackle the problem of unsupervised domain adaptation for
nuclei segmentation, where we have labeled source domain data and unlabeled
target domain data coming from different cancer types. Our labeled source data
has $N_{s}$ images $(x_{s},y_{s})$ and our unlabeled target domain has $N_{t}$
images $(x_{t})$.

###  3.2 Segmentation Loss and Mutual Information Maximization

We use a combination of dice loss and binary cross-entropy loss for supervised
segmentation training $(L_{seg})$. For mutual information (MI) maximization,
we use Jensen-Shannon Divergence (JSD)-based lower bound proposed in [10].
This bound allows us to estimate the MI with just one negative example for
each positive example. In our framework, we define nuclei pixels in the source
domain with nuclei pixels in the target domain as positive pairs while
background pixels in the source domain with nuclei pixels in the target domain
as a negative pair.

We use a simple $1\times 1$ convolution network with the same number of input
and output channels, followed by batch normalization and ReLU as the
projection head. The output of the backbone network is passed to the
projection head for generating representation for MI loss. Projected feature
representation and segmentation labels are used for MI maximization. Since
labels are not available for the target domain, we use a segmentation network
output from the same iteration as pseudo-labels. Typically, we would like to
contrast all the positive pair pixels with negative pair pixels. However, for
computational feasibility, we mean pool the projected representations using
the segmentation labels to get aggregated background and nuclei pixel
representations. Further, mean pooled source nuclei representation with mean
pooled target nuclei representation is treated as a positive pair, whereas
mean pooled background pixel representation with mean pooled target nuclei
representation is treated as a negative pair. We define our JSD estimator as:

$\hat{I}_{\omega}^{JSD}(Z_{s};Z_{t}):=\mathbb{E}_{P(Z_{s},Z_{t})}[-sp(-T_{\omega}(z_{s},z_{t}))]-\mathbb{E}_{P(Z_{s})P(Z_{t})}[sp(T_{\omega}(z_{s}^{\prime},z_{t}))]$
(1)

where $z_{s}$ is mean pooled source nuclei representation, $z_{t}$ is mean
pooled target nuclei representation, $z_{s}^{\prime}$ is mean pooled
background representation for the source sample, and $sp(z)=log(1+e^{z})$ is
the softplus function. Here, $T_{\omega}:Z_{s}\times Z_{t}\to\mathbb{R}$ is a
discriminator network with trainable parameters $\omega$ which are jointly
optimized to distinguish between a paired-sample from a joint distribution
(positive pair) and a paired sample from product of marginals (negative pair).
For discriminator network, we use the concat architecture proposed in [10].

Figure 1: Images pass through the backbone network and segmentation head for
the segmentation training and projection head for contrastive training. Source
image actual label is used for segmentation loss and to obtain the feature
representation of nuclei and background area. Target image prediction is used
as the pseudo label to get the feature representation of the nuclei area.
Masked features are average pooled for generating positive and negative pairs
for mutual information maximization.

###  3.3 Training

As shown in Figure 1, our architecture is divided into 4 parts - 1) backbone
encoder-decoder network ($\theta_{b}$), 2) segmentation head ($\theta_{s}$)
for generating segmentation masks, 3) projection head ($\theta_{p}$) for
generating feature representations for MI maximization and 4) discriminator
network $(T_{\omega})$ for estimating MI.

We divide our training into 2 steps. In step 1, for warming up the model and
achieving reasonable pseudo-labels for target data, we pretrain the backbone
network ($\theta_{b}$) and segmentation head ($\theta_{s}$) using source data
on segmentation loss.

$(\hat{\theta_{b}},\hat{\theta_{s}})=\operatorname*{argmin}_{\theta_{b},\theta_{s}}\frac{1}{\left|X_{s}\right|}\sum_{(x_{s},y_{s})}L_{seg}(y_{s},\hat{y_{s}})$
(2)

In step 2, we continue the training the model with the MI loss between labeled
source data points and pseudo-labeled target data points.

$\begin{split}(\hat{\theta_{b}},\hat{\theta_{s}},\hat{\theta_{p}},\omega)=\operatorname*{argmin}_{\theta_{b},\theta_{s}}\frac{1}{\left|X_{s}\right|}\sum_{(x_{s},y_{s})}L_{seg}(y_{s},\hat{y_{s}})+\\\
\operatorname*{argmax}_{\omega,\theta_{p},\theta_{b},\theta_{s}}\frac{1}{\left|X_{pair}\right|}\sum_{(x_{s},y_{s}),(x_{t},\hat{y_{t}})}\hat{I}_{\omega}^{JSD}(Z_{s};Z_{t})\end{split}$
(3)

where $Z_{s}$ and $Z_{t}$ are mean pooled nuclei representation for source and
target respectively, and $X_{pair}$ define randomly paired source and target
image for MI maximization.

##  4 Experiments

###  4.1 Dataset and Implementation

We evaluate our approach using the similar setting used in [15] for Nuclei
Semantic Segmentation and [27] for Nuclei Instance Segmentation. In [15],
authors used the Kidney Renal Clear Cell Carcinoma (KIRC) dataset [14], Triple
Negative Breast Cancer Cell (TNBC) dataset [18], and Stomach adenocarcinoma
(STAD) [12] of the TCIA repository. There are $486$ images of $400\times 400$
pixel size in KIRC, $50$ images of $512\times 512$ pixel size in TNBC, and
$99$ images of $256\times 256$ pixel size in STAD. [15] evaluated UDA Nuclei
Semantic Segmentation for following cancer type domain shift - TNBC to KIRC/
TCIA, and TCIA to KIRC/ TNBC. Following the same setup for comparison, we use
standard U-Net architecture as the backbone encoder-decoder network and $80\%$
of images for training, $10\%$ for validation, and $10\%$ for testing. We
report average dice scores from five runs with different splits for accounting
for sampling bias.

In [27], the authors used colorectal nuclear segmentation and phenotype
dataset (CoNSep) [7] as the source domain and PanNuke dataset [5] comprising
19 cancer types as the target domain for UDA in nuclei instance segmentation
and classification task. As used in their paper, we use labeled CoNSep and the
first split of PanNuke for training, second split for validation, and third
split for testing. We focused on the instance segmentation task and didn’t use
classification labels for training. We use HoverNet as the backbone encoder-
decoder network with MI maximization module included in the nuclei
segmentation branch. We report Dice Score, Aggregate Jaccard Index (AJI),
Panoptic Quality (PQ), Detection Quality (DQ), and Segmentation Quality (SQ)
for comparison.

All experiments are carried out in PyTorch using $1$ A100 GPU for nuclei
semantic segmentation and $4$ A100 GPUs for nuclei instance segmentation. For
the nuclei semantic segmentation experiment, we train the model for $1000$
iterations using only segmentation loss and $9000$ iterations with both
segmentation and MI maximization loss. We use a batch size of $4$, Adam
optimizer with a learning rate of $1e-3$, and rotation augmentation. We
randomly pair labeled source images with unlabeled target images during the MI
maximization step. For nuclei instance segmentation experiment, we follow the
training details provided in [27] and [7]. We extend our approach only for
nuclei instance segmentation and will focus on classification in our future
work. First, we freeze the encoder and train the decoder for $50$ epochs, with
the first $10$ epochs only using segmentation loss and the subsequent $40$
epochs using both segmentation and MI loss. Then we unfreeze and train the
whole network for $50$ epochs with both losses. We use a batch size of $16$
and Adam optimizer with a learning rate of $1e-4$.

###  4.2 Evaluation

Table 1: Nuclei Semantic Segmentation Results for UDA. Source Domain | TNBC | TCIA
---|---|---
Target Domain | KIRC | TCIA | KIRC | TNBC
Source Only | 0.713 | 0.680 | 0.710 | 0.791
DA_ADV [4] | 0.726 | 0.734 | 0.708 | 0.787
CellSegUDA [9] | 0.728 | 0.765 | 0.705 | 0.805
SelfEnsemblingUDA [15] | 0.727 | 0.761 | 0.715 | 0.816
MaNi | 0.733 | 0.776 | 0.727 | 0.821

####  4.2.1 Nuclei Semantic Segmentation

\- Table 1 compares our approach with recently proposed approaches for UDA for
nuclei semantic segmentation 111Other approach results reported using best
self-implementation. MaNi outperforms all other approaches for both TNBC to
KIRC/ TCIA shift and TCIA to KIRC/ TNBC shift. The next best performing
approach proposed in [15] uses multiple components consisting of a
discriminator and a reconstruction network, consistency regularization between
perturbed target images, and post-processing with CRF. While MaNi only uses
the MI maximization module with segmentation loss for training, reducing the
complexity of the architecture. We attribute high gains to the MI module’s
ability to transfer nuclei segmentation knowledge from labeled source pixels
to pseudo-labeled target pixels.

Table 2: Nuclei Instance Segmentation Results for UDA from CoNSep to PanNuke. Method | Dice | AJI | DQ | SQ | PQ
---|---|---|---|---|---
Source Only | 0.576 | 0.387 | 0.461 | 0.657 | 0.342
GRL [6] | 0.723 | 0.509 | 0.587 | 0.756 | 0.450
Paul et al. [19] | 0.731 | 0.501 | 0.600 | 0.751 | 0.446
Yang et al. [27] | 0.740 | 0.516 | 0.602 | 0.753 | 0.460
MaNi | 0.735 | 0.534 | 0.621 | 0.742 | 0.477

####  4.2.2 Nuclei Instance Segmentation

\- Results for UDA in nuclei instance segmentation from CoNSep data to PanNuke
data is reported in Table 2222Other approach results reported from [27]. MaNi
performs competitively to the best performing approach. MaNi demonstrates
superior performance over PQ, DQ, and AJI, highlighting that our approach can
detect and segment more nuclei accurately and suffers less from the issue of
false negatives. While in terms of SQ and Dice Score, MaNi is competitive to
other approaches.

Table 3: Impact of MI loss weight. Source Domain | TNBC | TCIA
---|---|---
Target Domain | TCIA | TNBC
1 | 0.776 | 0.821
0.1 | 0.769 | 0.765
0.01 | 0.751 | 0.765

Table 4: Impact of type of MI loss. Source Domain | TNBC | TCIA
---|---|---
Target Domain | TCIA | TNBC
Mean-Pooling | 0.776 | 0.821
Max-Pooling | 0.751 | 0.817
Random Pixels | 0.771 | 0.809

###  4.3 Ablation Studies

We study the impact of different weightage on the MI loss for training. As
shown in Table $3$, the default choice of $1$ performs best. The impact of MI
loss dilutes with reduction in weight, leading to a drop in performance.
Further, we study how replacing the average pooling approach with the max-
pooling approach or sampling random pixels approach (N=$4$), as done in [3],
impacts performance. Average pooling performs superior to other approaches.

##  5 Conclusion

In this paper, we propose a JSD-based MI loss for UDA for Nuclei Semantic
Segmentation and Instance Segmentation. We perform comprehensive experiments
with different architecture - UNet and HoverNet and for different cancer-type
domain shifts - TNBC to TCIA/KIRC, TCIA to KIRC/TNBC, and CoNSep to PanNuke.
We highlight that our method leads to gain for both semantic segmentation and
instance segmentation. We plan to extend this approach for nuclei
classification tasks and subsequently to general imaging tasks in our future
work.

####  5.0.1 Acknowledgements

This work was supported by NIDDK of the National Institutes of Health under
award number K23DK117061-01A1 and Litwin IBD Pioneers Award of the Crohn’s &
Colitis Foundation.

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