# Engineering holographic flat fermionic bands

Nicolás Grandi<EMAIL_ADDRESS>Instituto de Física de La Plata -
CONICET, C.C. 67, 1900 La Plata, Argentina.  Departamento de Física - UNLP,
Calle 49 y 115 s/n, 1900 La Plata, Argentina.  Vladimir Juričić
<EMAIL_ADDRESS>Departamento de Física, Universidad Técnica Federico Santa
María, Casilla 110, Valparaíso, Chile.  Nordita, KTH Royal Institute of
Technology and Stockholm University, Hannes Alfvéns väg 12, 106 91 Stockholm,
Sweden  Ignacio Salazar Landea<EMAIL_ADDRESS>Instituto de Física de La
Plata - CONICET, C.C. 67, 1900 La Plata, Argentina.  Rodrigo Soto-Garrido
<EMAIL_ADDRESS>Facultad de Física, Pontificia Universidad Católica de
Chile, Vicuña Mackenna 4860, Santiago, Chile

###### Abstract

In electronic systems with flat bands, such as twisted bilayer graphene,
interaction effects govern the structure of the phase diagram. In this paper,
we show that a strongly interacting system featuring fermionic flat bands can
be engineered using the holographic duality. In particular, we find that in
the holographic nematic phase, two bulk Dirac cones separated in momentum
space at low temperature, approach each other as the temperature increases.
They eventually collide at a critical temperature yielding a flattened band
with a quadratic dispersion. On the other hand, in the symmetric (Lifshitz)
phase, this quadratic dispersion relation holds for any finite temperature. We
therefore obtain a first holographic, strong-coupling realization of a
topological phase transition where two Berry monopoles of charge one merge
into a single one with charge two, which may be relevant for two- and three-
dimensional topological semimetals.

_Introduction._ Flat electronic bands have recently attracted significant
attention due to the experimental realization of twisted bilayer graphene [1,
2]. They naturally promote interaction effects as dominant, leading to a rich
landscape of possible strongly correlated phases [3]. Strongly interacting
systems, however, may be prohibitively difficult to address within the
traditional frameworks, such as perturbation theory. Furthermore, numerical
methods are restricted to exact diagonalization due to the sign problem
present in systems involving fermionic degrees of freedom.

Given the necessity of using a nonperturbative approach, the AdS/CFT
correspondence, also known as holographic duality [4, 5, 6], has been broadly
used to investigate strongly coupled systems. In the more general sense, this
duality proposes an equivalence between systems described by strongly
interacting quantum field theories and weakly coupled systems in a curved
spacetime with an extra spatial dimension. As such, these holographic methods
so far have been useful to gain qualitative insights on condensed matter
systems, for instance, non-Fermi liquids, high Tc superconductors, and
topological systems, among others [7, 8, 9]. Given that flat bands favor the
effects of the strong electronic interactions, including the ensuing new
quantum phases and phase transitions, it is then rather natural to apply the
holographic duality to address the physics at strong coupling therein.

In this context, it is worthwhile emphasizing that a Lifshitz geometry may be
the natural setting for the construction of the holographic flat bands, given
that, at least from a perturbative point of view, the scaling of the
electronic density of states [$\rho(\varepsilon)$] with energy ($\varepsilon$)
in $d$ spatial dimensions is tunable by the dynamical exponent ($z$),
$\rho(\varepsilon)\sim|\varepsilon|^{\frac{d}{z}-1}$. It is then expected that
a tunable dynamical exponent could yield an instability towards a symmetry
broken phase, as in the model for multi-Weyl semimetal we have recently
proposed [10], realizing a phase transition into a nematic phase, as also
found in Ref. [11]. However, none of these holographic models feature the
fermions in the bulk geometry that may in turn explicitly yield the
holographic flat fermionic bands.

Here, we address this important problem by incorporating the fermions directly
into the model of Ref. [10]. By computing the Green function in the background
geometry, we show that the dispersion relation of the fermions indeed displays
the flattening feature. In particular, we find that in the nematic phase, two
Dirac cones separated in momentum space at low temperature, approach each
other as the temperature increases. Eventually, at the critical temperature,
they collide yielding a flattened band with a quadratic dispersion relation.
On the other hand, in the symmetric (Lifshitz) phase, this quadratic
dispersion relation holds for any finite temperature. Therefore, besides the
holographic band flattening, we find a first holographic, strong-coupling
realization of a topological phase transition where two Berry monopoles of
charge one merge into a single monopole with charge two, which pertains to
topological metals in both two [12, 13, 14] and three [15, 16, 17, 18] spatial
dimensions.

_Free model._ Let us first discuss the flat bands within a $(2+1)$-dimensional
free Dirac fermion model. Our aim is to extract from it the symmetry breaking
pattern that we will later export into the holographic realm. It is defined by
the Hamiltonian

$H_{D}^{\prime}\\!=\\!-\gamma^{t}(\gamma^{x}p_{x}+\gamma^{y}p_{y})\otimes\mathbb{1}_{\mbox{\scriptsize$2\\!\times\\!2$\normalsize}}+i\,{m_{*}}\\!\left(\gamma^{x}\\!\otimes\\!\sigma_{2}-\gamma^{y}\\!\otimes\\!\sigma_{1}\right),$
(1)

where $\gamma^{\mu}=(\sigma_{3},-i\sigma_{2},i\sigma_{1})$ are the Dirac gamma
matrices, and the Pauli matrices $\sigma_{i}$ in the second term act on an
internal flavour index. This is analogous to the construction of the effective
Hamiltonian for bilayer graphene by coupling two single layers each containing
a linearly dispersing Dirac fermion. The spectrum reads

$\omega=\pm m_{*}\pm\sqrt{p_{x}^{2}+p_{y}^{2}+m_{*}^{2}}$ (2)

with gapped conduction and valence bands corresponding to both positive or
both negative signs respectively, while otherwise the valence and conduction
bands cross at zero energy. In the latter case, we obtain a low-energy
($p_{x}^{2}+p_{y}^{2}\ll m_{*}^{2}$) quadratic dispersion

$\omega\approx\pm\frac{1}{2m_{*}}\left(p_{x}^{2}+p_{y}^{2}\right),$ (3)

which still preserves rotational invariance.

The wave equation resulting from the Hamiltonian in Eq. (1) can be obtained
from the action

$S=S_{\sf free}-i\int d^{3}x\,\bar{\Psi}\not{W}\Psi.$ (4)

Here the first term is the free Dirac action for a pair of two-component
spinors, while the deformation corresponds to a coupling of the pair to a
constant non-Abelian vector field $W=m_{*}(\sigma^{1}dx+\sigma^{2}dy)$. This
explicitly breaks the $U(2)$ symmetry down to $U(1)$. Notice that spatial
rotational invariance is also broken. However, there is a “mixed” rotational
symmetry preserved in the system which is realized by compensating a spatial
rotation with an internal transformation generated by $\sigma_{3}$. The latter
is enough to preserve the rotational invariance of the spectrum.

_Holographic model._ We now construct a bottom up holographic dual to a
strongly coupled version of the model in Eq. (4). To do so, we first extend
the global boundary symmetries to the bulk as gauge symmetries, introducing
the relevant bulk gauge fields. Gauged space-time symmetries require a
dynamical metric, minimally described by Einstein gravity. We include a
negative cosmological constant in order to get AdS asymptotics. Moreover, we
add the corresponding Yang-Mills fields needed to gauge the boundary $U(2)$
symmetry

$S=\int
d^{4}x\sqrt{-g}\left(R-2\Lambda\right)-\frac{1}{4}\int\left[F\wedge{{}^{\\!\star}F}+\mathrm{Tr}\left(G\wedge{{}^{\\!\star}G}\right)\right]\,,$
(5)

where ${F}=dA$ represents a $U(1)$ gauge field strength accounting for the
conservation of particle number, while $G=d{B}-i(q/2){B}\wedge B$ is the
strength of a $SU(2)$ gauge field, accounting for the flavor symmetry in UV of
the model in Eq. (4). One may therefore roughly state that, at this step,
matter is hidden behind the black hole horizon, and an exterior observer can
only see the total charges.

Hence, we look for asymptotically AdS black holes solutions, with the generic
_ansatz_ for the metric

$\mathrm{d}s^{2}=\frac{1}{r^{2}}\Big{(}\\!-\\!Nf\mathrm{d}t^{2}+\frac{dr^{2}}{f}+dx^{2}+dy^{2}+2h\,dx\,dy\Big{)}\,,$
(6)

in terms of purely $r-$dependent functions $f,N$ and $h$, which close to the
boundary satisfy $f,N\to 1$ and $h\to 0$. To describe a boundary system at
finite temperature, we focus on black hole solutions with a horizon at finite
$r=r_{h}$, at which $N$ and $h$ are bounded, and $f$ vanishes linearly with
$f^{\prime}={4\pi T}/{\sqrt{N}}$.

For the gauge fields, we write

$\displaystyle~{}\,\,\,{A}$ $\displaystyle=0$ (7) $\displaystyle B$
$\displaystyle=\frac{1}{2}\left(Q_{1}\sigma_{1}\,+Q_{2}\sigma_{2}\,\right)dx+\frac{1}{2}\left(Q_{1}\sigma_{2}+Q_{2}\sigma_{1}\right)dy\,.$

The gauge field $A$, coupled at the boundary to the particle current, is
turned off, implying that the chemical potential vanishes. Here $Q_{1,2}$ are
$r$-dependent functions which are finite at the horizon. On the boundary, we
turn on a deformation analogous to the second term of the action (4), by
requiring $B\to W$ or equivalently $Q_{1}\to 2m_{*}$. This reproduces the
symmetry breaking pattern of our free model in the present holographic (i.e.
strongly coupled) setup. This approach has already been used to construct
holographic duals of fermions with particular dispersion relations, as, for
instance, Weyl semimetals [19], multi-Weyl semimetals [11, 20], nodal line
semimetals [21], and Weyl $Z_{2}$ semimetals [22].

The resulting phase diagram in terms of the gauge coupling $q$ and the
dimensionless temperature $T/m_{*}$ is shown in Fig. 1. At high enough
$T/m_{*}$ and $q$, only rotational invariant solutions are present, with $h$
and $Q_{2}$ equal exactly zero. The model flows towards a metastable Lifshitz
geometry in the IR, with a scaling exponent $z$ that is completely fixed by
the coupling $q$ as the root of a cubic polynomial [23]

$q^{2}(z^{3}+z^{2})+(q^{2}-24)z-3(q^{2}+8)=0\,.$ (8)

As we lower the temperature or the gauge coupling, a boundary nematic phase
develops, since the fields $Q_{2}$ and $h$ in the bulk spontaneously acquire
non-vanishing expectation values. Full $z=1$ rotational invariance is however
recovered in the deep IR.

Figure 1: Phase diagram of the model in Eq. (4), showing the nematic phase
close to the origin and the Lifshitz one away from it. At the top, we see the
fermionic dispersion relation for infinite $T/m_{*}$, the dotted lines
corresponding to the approximation in Eq. (17). On the right hand side, the
dispersion corresponding to the Lifshitz phase for $T/m_{*}=47.74$, with the
lines of approximation (18) is displayed. At the bottom, it is shown the
anisotropic dispersion relation in the nematic phase, just below the critical
temperature $T/m_{*}\approx 0.075$, along the two directions of Fig. 2. We
here set $q=q_{D}=2$, i.e. $z=2.48$.

_Fermions in the bulk._ To confirm that our holographic model actually
describes flat bands, we need to obtain its fermionic dispersion relation. In
holography, boundary matter fluctuations are represented by the fermionic
degrees of freedom $\Psi$ in the bulk, satisfying the Dirac equation [24]

$\Gamma^{A}\,D_{A}\Psi=0\,.$ (9)

Here $\Gamma^{A}$ with $A\in\\{0,1,2,3\\}$ are the $(3+1)$-dimensional flat
space Dirac gamma matrices, and $D_{A}$ is the curved space and gauge
covariant derivative. The field $\Psi$ is an $SU(2)$ doublet of
$(3+1)-$dimensional spinors, containing a total of eight independent
components.

As our model is closely related to the non-Abelian $p$-wave superconductor
[25, 26, 27, 28, 29, 30], here we only sketch the construction of $D^{A}$. The
key step is to write the metric in the tetrad formalism

$ds^{2}=\eta_{AB}e^{A}e^{B}\,,$ (10)

where $\eta_{AB}=\text{diag}(-1,1,1,1)$ is the Minkowski metric and $e^{A}$
are the coframe fields, given by

$\displaystyle
e^{1}\equiv\displaystyle\frac{1}{2r}\left[\left(\sqrt{1\\!+\\!h}+\sqrt{1\\!-\\!h}\right)dx+\left(\sqrt{1\\!+\\!h}-\sqrt{1\\!-\\!h}\right)dy\right]\,,$
$\displaystyle
e^{2}\equiv\displaystyle\frac{1}{2r}\left[\left(\sqrt{1\\!+\\!h}-\sqrt{1\\!-\\!h}\right)dx+\left(\sqrt{1\\!+\\!h}+\sqrt{1\\!-\\!h}\right)dy\right]\,,$
$\displaystyle e^{3}\equiv\displaystyle\frac{1}{r\sqrt{f}}\;dr\,,\qquad\quad
e^{0}\equiv\displaystyle\frac{1}{r}\sqrt{Nf}\;dt\,.$ (11)

The corresponding spin connection can be then found through the torsionless
condition, for details and conventions see [31].

We furthermore conveniently perform Fourier transform from the transverse
coordinates $(t,x,y)$ to $(\omega,k_{x},k_{y})$, and re-scale the spinors as

$\Psi={r^{3/2}}(fN(1-h^{2}))^{-1/4}\psi_{k}(r)\,.$ (12)

Projecting into the $2+1$ transverse spacetime by considering eigenvectors of
the radial gamma matrix

$\psi_{k}=\left(\begin{array}[]{c}\psi_{+}\\\ \psi_{-}\end{array}\right)\,,$
(13)

we obtain the explicit form of the equations for the two components
$\psi_{\pm}$,

$\displaystyle\psi^{\prime}_{-}-\frac{i}{\sqrt{2f(1-h^{2})\tilde{h}}}U\cdot\psi_{+}=0\,,$
$\displaystyle\psi^{\prime}_{+}+\frac{i}{\sqrt{2f(1-h^{2})\tilde{h}}}U\cdot\psi_{-}=0\,,$
(14)

with $\tilde{h}\equiv 1+\sqrt{1-h^{2}}$, $H\equiv h+i\tilde{h}$ and

$U=\begin{pmatrix}\tilde{h}k_{y}-hk_{x}&\tilde{h}k_{x}-hk_{y}+\sqrt{\frac{2(1-h^{2})\tilde{h}}{fN}}\omega&\frac{1}{2}q_{D}\left(HQ_{1}-iH^{*}Q_{2}\right)&\frac{1}{2}q_{D}\left(HQ_{2}-H^{*}Q_{1}\right)\\\
\tilde{h}k_{x}-hk_{y}+\sqrt{\frac{2(1-h^{2})\tilde{h}}{fN}}\omega&hk_{x}-\tilde{h}k_{y}&\frac{1}{2}q_{D}\left(HQ_{2}-iH^{*}Q_{1}\right)&\frac{1}{2}q_{D}\left(H^{*}Q_{1}+iH^{*}Q_{2}\right)\\\
\frac{1}{2}q_{D}\left(H^{*}Q_{1}+iHQ_{2}\right)&\frac{1}{2}q_{D}\left(-iH^{*}Q_{1}-H^{*}Q_{2}\right)&\tilde{h}k_{y}-hk_{x}&\tilde{h}k_{x}-hk_{y}+\sqrt{\frac{2(1-h^{2})\tilde{h}}{fN}}\omega\\\
\frac{1}{2}q_{D}\left(-iHQ_{1}+H^{*}Q_{2}\right)&\frac{1}{2}q_{D}\left(-H^{*}Q_{1}-iHQ_{2}\right)&\tilde{h}k_{x}-hk_{y}-\sqrt{\frac{2(1-h^{2})\tilde{h}}{fN}}&hk_{x}-\tilde{h}k_{y}\end{pmatrix}.$
(15)

Close to the boundary, the fields $\psi_{\pm}$ tend to a constant
$\psi_{\pm}\to\psi_{\pm}^{UV}$. The four components of $\psi_{+}^{UV}$ are
interpreted as the expectation values of the dual fermionic operator, while
those of $\psi_{-}^{UV}$ are instead proportional to the external sources. On
the other hand, close to the black hole horizon, in-going boundary conditions
yield

$\psi_{\pm}=\psi_{h\pm}(r-r_{h})^{-\frac{i\omega}{4\pi T}}$ (16)

with $\psi_{h+}=-i(I_{2\times 2}\otimes\sigma_{2})\cdot\psi_{h-}$. This
constraint implies that $\psi_{\pm}^{UV}$ are not all independent. Indeed,
since the equations are linear, we can assume the linear relations
$\psi_{+}^{UV}=M_{+}\psi_{h}$ and $\psi_{-}^{UV}=M_{-}\psi_{h}$, where the
matrices $M_{\pm}$ must be obtained by numerical integration. This in turn
implies $\psi_{+}^{UV}=M_{+}M_{-}^{-1}\psi_{-}^{UV}$, allowing us to identify
the retarded fermionic correlator as $G=M_{+}M_{-}^{-1}$. Then the poles of
the correlator can be simply read off from the zeros of the determinant of
$M_{-}$.

_Fermionic dispersion relation._ Focusing on the lowest lying poles of the
retarded correlator, we obtain the dispersion relation of the holographic
fermion. Starting at the infinite temperature limit, its form reads

$\omega\approx-i+v_{f}k-i\Gamma k^{2}+\dots$ (17)

featuring a purely imaginary gap and a real part scaling linearly with $k$, as
expected for a massless fermion. At large $k$ non-linear contributions coming
from the broken conformal symmetry have to be included. The corresponding
dispersions are depicted in the plots at the top of Fig. 1.

As the temperature is lowered, the fermionic dispersion relation (17) gets
modified to

$\omega\approx-i+ck^{2}-i\Sigma k^{4}+\dots$ (18)

in agreement with expectations based on the free model. This can be seen in
the plots at the center-right of Fig. 1 We emphasize that the dispersion
relation in Eq. (18) fits to a quadratic real part close enough to the origin.
This behavior holds until the system enters the instability towards the
nematic phase. The dispersion relation then becomes direction dependent. To
illustrate this behavior, two representative cuts on the ${\bf k}$ plane are
shown at the bottom plots of Fig. 1. We observe a rather notable breaking of
the rotational invariance. Interestingly, there is a zero frequency excitation
at some finite $k=k_{*}$ along one of the directions, close to which a
relativistic behavior shows up. ${\sf Re}\,\omega\approx v_{*}(k-k_{*})$. This
is a consequence of a $z=1$ fixed point, which corresponds to the deep IR of
the zero temperature limit in the nematic phase.

Additionally, in Fig. 2 we show a 3D plot with the position of the poles of
the two point function in the momentum space. We see explicitly that
rotational symmetry is broken while a discrete $Z_{2}$ symmetry around the
$k_{x}+k_{y}$ axis is preserved.

Figure 2:  Real (top) and imaginary (bottom) parts of the quasinormal
frequencies as a function of $\vec{k}=(k_{x},k_{y})$ in the nematic phase,
just below the critical temperature $T/m_{*}\approx 0.075$. The contour plots
on the right hand side correspond to the 3D plots on the left. The orange and
yellow lines denote the cuts which were plotted at the bottom of Fig. 1.

As we can see in Fig. 3 the real part of the zero momentum quasinormal
frequency quickly grows for the nematic phase, while the imaginary part drops
to zero. These gaps lift the degeneracy at low energies according to the
intuition that flat bands in interacting systems should be generically
unstable. On the other hand, the imaginary part drops to zero, making the
quasiparticles long lived as the temperature is lowered.

Figure 3: Real and imaginary part of the quasinormal frequencies at ${\bf
k}=0$ as a function of the temperature for the nematic phase.

_Conclusions and outlook._ In this paper, we realized a construction of
fermionic flat bands in the context of holography by directly including the
fermions in the bulk geometry within a standard bottom-up approach. We
consider a holographic dual of a free fermionic model that mimics two copies
of Dirac fermions hybridizing as in a graphene bilayer and yielding a
quadratic low-energy dispersion. Interestingly, we found that the holographic
fermions inherit this quadratic dispersion relation from the free model, while
the thermodynamics of the system points towards a different Lifshitz exponent.
This should not be surprising given that the holographic system is strongly
coupled and the critical exponents are indeed expected to renormalize.

We here point out that since a holographic system is intrinsically strongly
coupled, and, on the other hand, a flat electronic band is highly degenerate,
we expect that the holograhic flat band undergoes a phase transition towards a
phase where the degeneracy is lifted. This is in fact consistent with the
present holographic setup, since the Yang-Mills black holes, featured in the
model, are generically unstable. This fact is then reflected as a phase
transition towards a nematic phase. The concomitant rotational symmetry
breaking then leads to the splitting of the quadratically dispersing nodes
into two relativistic Dirac points that move apart as we lower the
temperature. As such, this process can be thought as a holographic realization
of the Berry monopole splitting at strong coupling.

We will conclude by discussing some future directions. First, one may consider
a UV completion of our bottom-up model where the relation between the bosonic
and fermionic sector arises from a stringy construction [32, 33, 34, 35].
Furthermore, the free fermionic construction can be extended to an arbitrary
integer $n$ with the dispersion $\omega\sim k^{n}$ by using $n$ fermionic
flavors and the generators of the spin-$(n-1)/2$ representation of the $SU(2)$
group [11]. It would be therefore interesting to explore holographic duals of
the models with higher-$n$ fermionic dispersions. Finally, one may consider
generalizations of these fermionic models to non-relativistic Goldstone bosons
[36, 37].

_Acknowledgments:_ This work was supported by ANID-SCIA-ANILLO ACT210100
(R.S.-G. and V.J), the Swedish Research Council Grant No. VR 2019-04735
(V.J.), Fondecyt (Chile) Grant No. 1200399 (R.S.-G.), the CONICET grants
PIP-2017-1109 and PUE 084 “Búsqueda de Nueva Física”, and by UNLP grants
PID-X791 (I.S.L. and N.E.G.).

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figure table

# Transmission grating arrays for the X-ray spectrometer on Arcus Probe

Ralf K. Heilmann  Space Nanotechnology Laboratory, MIT Kavli Institute for
Astrophysics and Space Research, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  Alexander R. Bruccoleri  Izentis, LLC, Cambridge, MA
02139, USA  James A. Gregory  MIT Lincoln Laboratory, Lexington, MA 02421, USA
Eric M. Gullikson  Lawrence Berkeley National Laboratory, Berkeley, CA 94720,
USA  Hans Moritz Günther  MIT Kavli Institute for Astrophysics and Space
Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Edward Hertz  Center for Astrophysics, Harvard-Smithsonian Astrophysical
Observatory, Cambridge, MA 02138, USA  Renee D. Lambert  MIT Lincoln
Laboratory, Lexington, MA 02421, USA  Douglas J. Young  MIT Lincoln
Laboratory, Lexington, MA 02421, USA  Mark L. Schattenburg  Space
Nanotechnology Laboratory, MIT Kavli Institute for Astrophysics and Space
Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

###### Abstract

The Arcus Probe mission concept has been submitted as an Astrophysics Probe
Explorer candidate. It features two co-aligned high-resolution grating
spectrometers: one for the soft x-ray band and one for the far UV. Together,
these instruments can provide unprecedented performance to address important
key questions about the structure and dynamics of our universe across a large
range of length scales. The X-ray Spectrometer (XRS) consists of four parallel
optical channels, each featuring an x-ray telescope with a fixed array of 216
lightweight, high-efficiency blazed transmission gratings, and two CCD readout
arrays. Average spectral resolving power $\lambda/\Delta\lambda>2,500$ ($\sim
3500$ expected) across the 12-50 Å band and combined effective area $>350$ cm2
($>470$ cm2 expected) near OVII wavelengths are predicted, based on the
measured x-ray performance of spectrometer prototypes and detailed ray trace
modeling. We describe the optical and structural design of the grating arrays,
from the macroscopic grating petals to the nanoscale gratings bars, grating
fabrication, alignment, and x-ray testing. Recent x-ray diffraction efficiency
results from chemically thinned grating bars are presented and show
performance above mission assumptions.

###### keywords:

Arcus, x-ray spectrometer, critical-angle transmission grating, high-
resolution x-ray spectroscopy

*Corresponding author: Ralf K. Heilmann<EMAIL_ADDRESS>
##  1 Introduction

In our quest to understand the structure of the universe around us we cannot
be satisfied with measuring how things are today, but we also must understand
the dynamics of our universe, how our Milky Way and other galaxies formed and
aggregated, and how they will develop into the future. Images and low
resolution spectra do not provide the required precision data that informs us
about the kinetics and temperatures of baryons that underpin not only much of
the visible universe, but also trace the small and large-scale structure and
dynamics of its hottest components as highly ionized plasma. Emission and
absorption in these plasmas occurs predominantly in the soft x-ray and FUV
bands, calling for high-resolution spectroscopy at these short wavelengths. To
quote the Astro2020 Decadal Review:[1] “Astronomy became astrophysics with the
first spectrum. Spectroscopy determines compositions, magnetic field strength,
space motion, rotation, multiplicity, planetary companions, surface structure,
and other important physical traits…In the next decade, spectroscopy will be
the dominant discovery tool for astronomy.”

Arcus Probe will carry the required state-of-the-art instruments that can
deliver the desired spectra in much higher resolution and much shorter time
than existing, aging x-ray observatories. Details of the main science cases
for the Arcus Probe Mission can be found in Ref. 2.

##  2 The Arcus Probe X-ray Spectrometer (XRS)

The X-ray Spectrometer (XRS) instrument consists of four parallel, almost
identical optical channels (OC). In each channel an array of grazing-incidence
silicon pore optics (SPO),[3] with the layout of a sub-apertured version of
the NewAthena design,[4] forms an x-ray telescope with 12 m focal length. Sub-
aperturing in azimuth for grazing-incidence mirrors leads to an anisotropic
telescope point-spread function (PSF) that is much narrower in one dimension
than a mirror array that is fully populated in azimuth.[5, 6]. Just downstream
of the x-ray mirrors follows a grating petal, carrying an array of 216
critical-angle transmission (CAT) gratings that covers the mirror array. CAT
gratings are blazed transmission gratings that disperse soft x rays into
several diffraction orders, with the strongest orders on just one side of the
image near twice the blaze angle of the gratings. The gratings are aligned
with their dispersion axes parallel to the narrow direction of the PSF,
minimizing the spectrometer line-spread function (LSF) in the dispersion
direction. Some soft and most harder x rays are not diffracted and form an
image at the telescope focus, where x-ray CCDs provide energy resolution.[7]
Effective area for harder x rays in $0^{\rm{th}}$ order peaks around 1400 cm2
near 1.8 keV, and is still above 300 cm2 at 6.7 keV (summed over all four
OCs).[8]

---
Figure 1:  Arcus Probe schematics. Left: Spacecraft with instruments stowed in
launch vehicle fairing. Middle: In space after solar panel deployment
deployment. The UV spectrometer is on the left, the XRS on the right. Right:
After XRS boom deployment.

The combination of optics and grating petal comprises an OC. The four OCs are
mounted to a common forward assembly (FA), which in turn is mounted to a
coilable boom that is stowed during launch and deploys/uncoils after orbit
insertion (see Fig. 1). The boom is connected on the other end to the rear
assembly (RA), which serves as the interface to the spacecraft and holds the
two CCD detector subsystem assemblies (DSA), plus other systems. Information
about an earlier version of the boom can be found in Ref. 9.

The gratings in each channel are placed along the surface of a tilted Rowland
torus that also contains the telescope focus. This guarantees that best focus
in the dispersion direction for the non-zero orders is achieved, and also lies
on the Rowland torus surface. One DSA (DSA-1) sits at the focus of optical
channels OC-1 and OC-2. DSA-2 collects the strong blazed orders from OC-1 and
OC-2. Due to the symmetries of the design, DSA-2 can simultaneously lie on the
differently tilted Rowland torus for OC-3 and OC-4 and their foci. The
gratings in the latter two channels are blazed in the opposite direction from
the gratings in the first two channels, creating strong, focused diffraction
orders on DSA-1 (see Fig. 2). Each OC is offset from the others by a few mm in
the cross-dispersion direction, leading to four quasi-(anti)-parallel spectra
collected by only two compact DSAs. This so-called double tilted Rowland torus
design combines dense stacking of sub-apertured, narrow-LSF optics in a
limited aperture, with a minimal number of readout CCDs. It is described in
detail in Refs. 10 and 11. The readout cameras are described in detail in a
separate paper of this Special Issue.[7]

---
Figure 2:  Layout of the XRS. Left: Angled view, showing the thermal
precollimators and SPO arrays for the four OCs on the left. The grating petals
are obscured behind the SPO arrays. An x ray is shown that enters the lower
left OC (OC-3) and either lands at the telescope focus (black line to DSA-2)
or gets diffracted into a blazed order (rainbow colors to DSA-1). Right: View
from the x-ray source into the XRS, showing the four OCs and the two DSAs.

A grating facet consist of a $32.5\times 32$ mm2 silicon membrane aligned and
bonded to a metal facet frame.[12] All grating facets for all channels have
the same design, are operationally identical and can be exchanged with each
other. Blazing is simply achieved through proper tilting relative to the
incident x rays. The grating facets are mounted to grating windows in groups
of 4-6. The windows are designed to cover the SPO beams with the minimum
number of gratings of the given size.

---
Figure 3:  Left: Two different views of an optical channel. The grating petal,
holding 40 grating windows with a total of 216 grating facets, follows the
shape of the Rowland torus. Middle: Prototype grating window with four
$32.5\times 32$ mm2 CAT grating facets. Right: Silicon grating membrane, held
by tweezers.

##  3 Predicted XRS Performance

The key performance parameters for the XRS are effective area
($A_{\mathrm{eff}}$) and resolving power ($R=\lambda/\Delta\lambda$), with
$\lambda$ being the x-ray wavelength and $\Delta\lambda$ the smallest
wavelength difference that can be resolved. Effective area is mainly
determined by the mirror effective area that feeds the grating array, grating
diffraction efficiency (DE) summed over the collected diffraction orders,
geometrical loss factors (x-ray blockage from support structures, gaps between
gratings and CCDs), filter transmission, and CCD quantum efficiency. Resolving
power to first order is given by the diffraction angle of a given diffraction
order, divided by the spectrometer LSF. CAT gratings are blazed transmission
gratings that have advantages in diffraction efficiency and resolving power in
the soft x-ray band compared to existing grating instruments on Chandra[13]
and XMM/Newton.[14] In order to estimate XRS performance we need to understand
the blazing principle underlying CAT grating design and the alignment
tolerances that affect both $A_{\mathrm{eff}}$ and $R$. Alignment tolerances
are derived from detailed ray tracing that is described elsewhere in this
Special Issue and for previous smaller versions of Arcus.[8, 15, 10, 16]

##  4 CAT grating principle and structural hierarchy

CAT gratings feature ultra-high aspect-ratio, freestanding grating bars with
nm-smooth sidewalls.[17] Each grating is inclined such that x rays of
wavelength $\lambda$ impinge on the grating bar sidewalls at graze angles
$\alpha$ below the critical angle for total external reflection $\alpha_{c}$
(see Fig. 4). The diffraction angle $\beta_{m}$ for the $m^{\mathrm{th}}$
diffraction order is given by the grating equation

${\frac{m\lambda}{p}}=\sin\alpha-\sin\beta_{m},$ (1)

where $p$ is the grating period. Diffraction orders near the direction of
specular reflection from the sidewalls show increased efficiency (i.e.,
blazing). The small critical angles for soft x rays on the order of 1-2
degrees demand high-aspect ratio grating bars in order to intercept all
incoming photons. Furthermore, the bars should be as thin as possible to
minimize blockage and absorption. The grating period cannot be too large
compared to the x-ray wavelength to obtain diffraction orders that can be
sorted by order using the energy resolution of Si-based detectors. The design
initially fabricated for Arcus has a grating period $p=200$ nm, grating bar
depth $d=4$ micrometers, and bar thickness $b\approx 60$ nm. Recently we
demonstrated $d>5.5$ $\mu$m.[18]. Blazing is most efficient when
$\tan\alpha\approx(p-b)/d$. (For $d=4$ $\mu$m this means $\alpha\approx 2$
deg.)

---
Figure 4:  Left: Schematic cross-section through a CAT grating of period $p$.
The $m^{\rm{th}}$ diffraction order occurs at an angle $\beta_{m}$ where the
path length difference between x rays from neighboring grating slots is
$m\lambda$. Shown is the case where $\beta_{m}$ coincides with the direction
of specular reflection from the grating bar sidewalls
($|\beta_{m}|=|\alpha|$), i.e., blazing in the $m^{\rm{th}}$ order. Right:
Scanning electron micrograph (SEM) of a cleaved CAT grating membrane showing
top, cross-section and sidewall views of the 200 nm-period silicon grating
bars and their monolithically integrated 5 $\mu$m-period L1 cross supports (x
rays enter from the top and leave out the bottom).

CAT grating bars are not supported by a membrane, but freestanding. As seen on
the right in Fig. 4, the bars are held in place by a monolithically integrated
5 $\mu$m-period Level 1 (L1) support mesh. Additional support structures are
needed for the few-$\mu$m thin grating layer in order to manufacture large
enough CAT gratings that can cover large areas on the order of thousands of
square centimeters with a manageable number of gratings. Fig. 5 shows the
additional, much thicker and stronger Level 2 (L2) hexagonal support structure
on the scale of $\sim 1$ mm.

---
Figure 5:  Schematic showing the structural hierarchy of a CAT grating
membrane (not to scale). See text for details.

##  5 CAT Grating Fabrication

CAT grating fabrication has been described extensively in previous work and is
briefly summarized here.[19, 17, 20, 21] CAT gratings are currently made from
200 mm silicon-on-insulator (SOI) wafers. The thin Si device (“front side”)
layer of the SOI wafer is manufactured to thickness $d$ (see Fig. 4). Using
193 nm 4X optical projection lithography (OPL) at MIT Lincoln Laboratory,
patterns for CAT gratings, L1 and L2 structures are simultaneously transferred
into a silicon oxide layer that serves as a mask for the subsequent deep
reactive-ion etch (DRIE). The $\sim 0.6$ mm thick handle (“back side”) layer
of the SOI wafer is then DRIE’d with a hexagonal pattern that is aligned with
the front side L2 hexagons. The grating bars are aligned parallel to the
vertical {111} planes of the (110) device layer. Since DRIE leaves rough
sidewalls, this crystal orientation is used to “polish” the grating bar
sidewalls post-DRIE through immersion in KOH solution.[17] The gratings have
to be dried in a critical-point dryer to prevent stiction due to liquid-vapor
surface tension. Finally, the buried oxide layer separating device and handle
layers is removed in the areas not covered by Si, resulting in a monolithic Si
grating layer with freestanding CAT grating bars, integrated L1 and L2
supports, and a bulky L2 mesh.

CAT gratings can be fabricated in volume manufacturing mode from 200 mm SOI
wafers,[21] in principle allowing for $\sim 16-20$ gratings to be produced
from a single wafer.

##  6 CAT Grating X-ray Performance

The main performance requirements for x-ray diffraction gratings are high
diffraction efficiency and enabling high resolving power in a spectrometer
instrument.

###  6.1 Diffraction Efficiency

Diffraction efficiency has been measured repeatedly at beamline 6.3.2 of the
Advanced Light Source at Lawrence Berkeley National Laboratory. The focused,
tunable monochromatic x-ray beam has small enough diameter to be placed within
a single L2 hexagon, but it integrates over many tens of L1 periods. A slit-
covered photodiode detector is placed at the angle of a transmitted
diffraction order, and the grating is rotated to measure DE over a range of
several degrees in incidence angle. The detector remains in place during
grating rotation, since the change in diffraction angle as a function of
grating rotation is negligible. This is a major advantage of the transmission
geometry, since this also means that certain grating alignment tolerances are
very relaxed compared to reflection gratings.

The earliest x-ray testable freestanding CAT grating structures had larger
period ($p=574$ nm) and bulky L1 cross supports that were $>>1$ $\mu$m
wide.[22, 23] DE can be modeled using rigorous coupled-wave analysis
(RCWA).[24] Typically, CAT gratings perform in the range of 80-100% of RCWA
predictions for DE, where the difference from 100% can often be modeled with a
Debye-Waller-type (DW) roughness factor.[21]. Current prototypes with 200 nm
period, $d=4$ $\mu$m, and $b\approx 60-70$ nm deliver DE $>20$% in individual
orders and $>30$% when summed over the strongest (“blazed”) orders
($\beta_{m}\sim 2\alpha$) near O-K wavelengths (see left of Fig. 6). These
numbers include losses caused by absorption from the L1 supports.

Several 4 $\mu$m-deep gratings were measured at several wavelengths, and the
data is modeled with RCWA and a DW factor (see right of Fig. 6). The model is
then used to extrapolate efficiency between the measured wavelengths and
angles as an input for ray-trace-based effective area predictions for the XRS.
Ray tracing also takes into account alignment tolerances and the changing
efficiencies as a function of grating angles.[25, 8]

---
Figure 6:  Representative diffraction efficiencies of 4 $\mu$m-deep CAT
gratings from synchrotron data. Left: Example detector scan for $\lambda=2.5$
nm. DE is strongy blazed for $5^{\rm{th}}$ order at 21%. Summing over orders
4-7 gives a total DE of 32.2%. These values include absorption by L1 bars, but
not blockage by the L2 mesh. Right: Comparison of model efficiency (sum of
blazed orders collected by XRS readouts), multiplied by Debye-Waller-like
roughness factor, for 82% L1 open area fraction with synchrotron spot
measurements (sum of same blazed orders as for model) for a $32\times 32$ mm2
CAT grating[26]. The L2 mesh is not considered in this plot.

The synchrotron beam can be scanned over a whole grating to examine uniformity
of efficiency.[27, 28, 26, 12] Alternatively, we also measured efficiency in
the converging beam of an SPO, almost fully illuminating a large CAT grating
in an Arcus-like configuration, and verified that it agreed with synchrotron
spot measurements and exceeded model assumptions.[29]

###  6.2 Resolving Power

Resolving power $R$ is an expression of how much a spectral line is broadened
by the XRS response function. The main ingredients are the mirror LSF (PSF
projected onto the dispersion axis), the diffraction angles of the measured
orders, the distance of the gratings from focus, the aberrations of the
optical design, the CCD pixel size, and grating imperfections. Large $R$
requires large grating distance from focus and large diffraction angles, and
small values for the other terms.

The most obvious grating imperfection that would limit $R$ is a variation
$\Delta p$ of grating period $p$, since it would introduce a variation in
$\beta_{m}$ (see Eq. 1). For a Gaussian distribution of $\Delta p$ one can
define an effective grating resolving power $R_{g}=p/\Delta p$ as an
additional term in the instrument response function that contributes to the
broadening of measured spectral lines.[29] $R_{g}$ is an upper limit to the
resolving power of the XRS and can be measured in the following way: First,
the PSF of a focusing optic (the “direct beam”) is measured with a narrow-line
soft x-ray source, such as the well-characterized Al-Kα doublet. Then, a
grating is inserted into the beam, and the source spectrum is measured in the
highest accessible diffraction order. The measured spectrum is a convolution
of the known source spectrum with the measured LSF of the direct beam and a
Gaussian of width $\Delta\beta_{m}$ caused by $\Delta p$. Thus, $\Delta p$ and
$R$ can be extracted from fitting to the measured spectrum. This measurement
has been performed up to 18th order at the Marshall Space Flight Center Stray
Light Facility using a slumped glass mirror pair[28, 26, 30] and up to 21st
order at the PANTER x-ray facility using an SPO.[21, 29] In all cases
$R_{g}\approx 10^{4}$ was found, which significantly exceeds the $R_{g}>3850$
Arcus requirement.

---
Figure 7:  Spectrum of Al anode from two mutually aligned and simultaneously
illuminated CAT gratings, showing the Al-Kα doublet measured in
$18^{\mathrm{th}}$ order.[29] Camera pixels are 20 $\mu$m in size. The red
line is the natural line shape of the doublet,[30] and the dashed line is the
best fit to the data. The black solid line is the curve for the lower
$3\sigma$ confidence limit, corresponding to $R_{G}=8157$. The upper $3\sigma$
confidence limit includes $R=\infty$. The dash-dotted lines show the
individual K${}_{\alpha_{1}}$ and K${}_{\alpha_{2}}$ components with their
natural widths on top of the weak sloped background.

A linear array of four co-aligned CAT gratings, illuminated by a pair of
confocal SPOs in an Arcus-like setup, also performed as expected.[12]. As
shown in Fig. 7, we recently derived $R_{g}\approx 1.3^{+\infty}_{-0.5}\times
10^{4}$ ($3\sigma$) for a different pair of co-aligned gratings,
simultaneously illuminated by a single SPO with LSF $\sim 1$ arcsec (full
width half max - FWHM).[29]

The limiting factor in resolving power for the Arcus XRS therefore is not the
quality of CAT gratings, but the combined LSF from the whole OC SPO petal,
required to be $<3.4$ arcsec (FWHM), and expected to be 2.6 arcsec (FWHM).

###  6.3 XRS Figures of Merit

It is useful to discuss the leap in performance that CAT grating technology
provides in the soft x-ray band over previous x-ray grating technologies.

---
Figure 8:  Comparison of TG diffraction efficiencies and figure of merit.
Left: For Chandra gratings the sum of $\pm 1^{\mathrm{st}}$ order DEs from
calibration files is shown [31]. These DEs include all losses from support
structures and gaps between facets. For CAT gratings all orders within the
blaze envelope are summed up, and a total loss of 31% from support structures
and gaps is assumed. “4 $\mu$m CAT” and “6 $\mu$m CAT” are from synchrotron
measurements of CAT gratings with $d=4$ and $\sim 6$ $\mu$m. The latter are
from the first prototypes of this thickness.[18, 32] “CAT model” is the sum of
blazed order DEs from a model with $d=6$ $\mu$m, $b=40$ nm, and 1.5 nm DW
roughness parameter. Right: Figure of merit for weak line detection
($\sim\sqrt{A_{\mathrm{eff}}R}$). The same mirror effective area and constant
detector efficiency is assumed for all plots. For CAT gratings, $R=10^{4}$ is
assumed as an upper limit, based on conservative analysis of lab measurements
[28, 30, 29]. The HETG and LETG instruments complement each other over the
shown range. Chandra with CAT gratings would be superior by far for
$\lambda>1$ nm due to a combination of higher DE and blazing into higher
orders.

The High and Low Energy Transmission Grating Spectrometers (HETGS[13], $p=200$
(HEG) and 400 nm (MEG); and LETGS, [33] $p=991$ nm) on Chandra have the
advantage of small mirror LSF on the order of 0.5 arcsec, but most of the
diffracted photons land at small angles in $m=\pm 1^{\rm{st}}$ orders
($\beta_{m}\approx\lambda/p=1$-$25\times 10^{-3}$ for the 1-5 nm wavelength
band). For the Arcus XRS, CAT gratings are inclined by 1.8 degrees relative to
the incident x rays, and the strongest orders are found near
$\beta_{m}=0.063$, relatively independent of $\lambda$. Chandra gratings were
designed as phase-shifting gratings with high efficiencies below 1 nm, and
peaking near 0.7 nm wavelength. Fig. 8 compares diffraction efficiencies for
different grating types. CAT gratings are designed for broadband high
efficiency for wavelengths longer than 1 nm. Arcus assumes performance shown
as “4 um CAT”. The “6 um CAT” curve shows results from deeper, more recent
gratings with $d\sim 5.5$-6 $\mu$m that could be used instead, which would
improve Arcus performance further. The right side of Fig. 8 is an “apples-to-
apples” comparison, assuming CAT gratings were placed on Chandra. It shows a
figure of merit for the detection of absorption lines, which is proportional
to $\sqrt{A_{\rm{eff}}\times R}$. We conservatively limited $R$ to $10^{4}$,
even though the Chandra mirror LSF would allow for higher values of $R$. CAT
gratings outperform Chandra gratings significantly for $\lambda>1$ nm.

---
Figure 9:  Comparison of figures of merit between soft x-ray spectroscopy
instruments in operation and the Arcus Probe XRS. “Chandra Gratings” refers to
the better of HEG, MEG, and LETG. The curve for the Resolve microcalorimeter
on XRISM[34] assumes that the closed aperture door is open. The wavelength
ranges for important plasma diagnostics lines are also indicated. Left: Figure
of merit for absorption line detection. Right: Figure of merit for emission
line detection.

Fig. 9 compares two different figures-of-merit for actual instruments on
different missions with the predicted Arcus performance. Despite the wider LSF
compared to Chandra, the Arcus XRS is expected to exceed Chandra HETG and
XMM/Newton RGS figures of merit by 1-2 orders of magnitude across the 1-5 nm
band, due to a combination of CAT grating properties and the larger mirror
effective area. Also shown is the Resolve instrument on XRISM, which is a
microcalorimeter with fixed energy resolution of slightly below 5 eV. It
provides better performance at much shorter wavelengths than shown.[34]

##  7 CAT Grating Alignment

Alignment tolerances for CAT gratings are derived using ray-tracing models as
described in Refs. 8, 15, 10, 16. Positioning tolerances are in the 200 $\mu$m
to mm ($3\sigma$) range, well within precision machining capabilities. Two
rotational degrees of freedom require custom metrology for alignment before
the grating membrane is bonded to its metal facet frame:

Grating roll (rotation around the grating normal, which also rotates the
dispersion axis) can be measured using visible light diffraction from the L1
cross support mesh,[18] which is defined in the OPL mask to be oriented at 90
degrees from the CAT bars.[21] Before bonding, roll relative to the facet
frame sides is adjusted to be the same for all grating facets, using a
reference grating facet.

Grating yaw, which controls the angle of incidence onto the CAT grating bar
sidewalls, requires a separate measurement of the grating bar tilt relative to
the grating membrane surface normal using small-angle x-ray scattering
(SAXS).[35, 36] Using laser reflection, the measured bar tilt is then
compensated for during the bonding of the Si membrane to its metal frame, such
that the average grating bar direction ends up being parallel to the normal of
the facet frame bottom, which is the mechanical reference surface for mounting
to the grating window. The success of this alignment method within Arcus
tolerances was demonstrated with x rays for a $2\times 2$ grating window.[37,
29]

##  8 Increasing Diffraction Efficiency through Bar Thinning

Besides fabricating CAT gratings with $d>4$ $\mu$m and reducing the cross
sections of support structures, XRS effective area can also be improved by
reducing the grating bar width $b$. The fabrication method described above
typically results in bar widths in the 60 nm range. Smaller widths could in
principle be achieved by creating an oxide mask with lower duty cycle (the
ratio of mask line width to grating period). However, the resulting thinner
bars can suffer from increased damage and destruction during the more
aggressive wet etching and cleaning steps. Obviously, thicker bars are less
fragile and preferred during wet processing steps.

Grating bar width can be reduced after the above fabrication steps using
repeated cycles of oxidation - which consumes a small amount of Si - and oxide
removal using HF vapor. Both steps are gas-based and gentle. In a first set of
experiments we demonstrated bar thinning on gratings that were not
freestanding, simply by removing native oxide with HF vapor, letting the
native oxide layer reform in air, and repeating the cycle.[37] We subsequently
repeated the experiment on three freestanding gratings, after measuring their
DE. Thinning is observable in top and bottom SEM images after about 10 cycles.
Here we present first comparisons of DE before and after bar thinning.

Of the three tested gratings with $d\sim 6$ $\mu$m, one each underwent 10
(SP1), 20 (SP3) and 30 (SP5) cycles of HF vapor oxide removal and native oxide
regrowth. The oxide was allowed to reform for at least 24 hours before the
next HF vapor etch. Fig. 10 shows top-down SEM images of grating SP3 before
and after treatment. After image analysis we estimate the CAT grating bars to
be $\sim 4\pm 2$ nm thinner, changing from $\sim 57$ to $\sim 53$ nm at the
top. Changes in thickness $b$ deeper into the gratings obviously can not be
discerned from SEM images. For grating SP1 we can not detect any clear
changes. The changes for grating SP5 look similar to the ones for SP3.

---
Figure 10:  Top down scanning electron micrograph (SEM) images of grating SP3
before (left) and after 20 cycles of HF vapor etching and oxidation in ambient
air (right). The 200 nm-period grating bars are slightly thinner after 20
cycles, but the difference is difficult to quantify precisely from SEM images
alone.

The gratings were measured for diffraction efficiency at the synchrotron
eleven months after their initial synchrotron measurements, and about two
months after the start of the HF vapor etching cycles. Measurements were taken
inside the same L3 hexagon as before on each sample, except for SP1, which
unfortunately was damaged during handling in the area of the previously
measured hexagon. For SP1 another hexagon near the original location was
selected, and we could not discern a systematic increase or decrease of DE.
For gratings SP3 and SP5 we show DE comparisons in Fig. 11. One can clearly
see an increase in $0^{\rm{th}}$ order efficiency, especially at normal
incidence, which indicates less blockage by Si. For individual higher
diffraction orders the DE increase is less pronounced, but it is clearly
visible when summing over the blazed orders. For both gratings we find an
increase in blazed efficiency in the range of 2-3% in the angular range where
blazing is most efficienct. Preliminary modeling of the DE with RCWA indicates
that the CAT grating bars became $\sim$ 4-5 nm thinner on average, with
average $b$ approaching $\sim$ 42-44 nm. The data also show that DE is
significantly higher than assumed for Arcus (see Fig. 6).

While it is clear that HF vapor etching of silicon oxide leads to thinner
grating bars, the etch could also change the roughness of the sidewalls,
thereby increasing (lower roughness) or decreasing (higher roughness) DE. It
is unclear why 10 treatment cycles seem to have caused little change, and why
we see little difference in outcome between 20 and 30 etch/oxidation cycles.
We plan to perform more systematic experiments, including fast growth of
thicker oxide layers at elevated temperatures, to understand the different
trade-offs between experimental conditions and outcomes.

There will be a mechanical limit to making grating bars thinner, when the
grating membrane could get damaged by launch vibrations. Vibration and
temperature cycling testing so far have not revealed any problems,[26] but
will have to be repeated for thinner and thinner structures.

---
Figure 11:  Diffraction efficiencies at $\lambda=1.75$ nm for orders 0-10 and
sum of orders 3-9 as a function of incidence angle. Dashed lines are before
thinning, solid lines after thinning. Left: Grating SP3. Right: Grating SP5.

##  9 Future Work

Future improvements in DE are possible through thinner CAT grating bars and
thinner L1 and L2 support structures. The Arcus Probe XRS assumes an L1 duty
cycle of 18%, but large CAT gratings with 10% duty cycle have been fabricated
previously[38] and could be used for Arcus. The L2 open area fraction could be
increased from the current 81% to a higher value. Deeper gratings also show
higher DE, but the difference with 4 $\mu$m deep gratings is most pronounced
at smaller incidence angles, which are preferred for larger mission concepts
with smaller optics PSF, such as Lynx.[39, 40] All of these improvements are
actively being investigated.

##  10 Summary

Arcus Probe is a mission concept that features two high-resolution
spectrometers. The XRS described in this work is an instrument with 1-2 orders
of magnitude improved performance compared to existing missions in the 1-5 nm
soft x-ray band. The four parallel OCs provide large effective area and high
redundancy in a small footprint. Alignment tolerances are well-understood and
well within metrology and manufacturing capabilities. Co-alignment with the UV
spectrometer (UVS) is described in another paper in this Special Issue,[41] as
well as the UVS itself.[42]

SPOs and CAT gratings already perform at required levels or exceed them. A
volume manufacturing approach utilizing tools from the semiconductor and MEMS
industries has been developed that can produce the required 864 CAT gratings
efficiently in two years. Mechanical structures (facet frames, grating
windows, grating petals, etc.) are standard precision engineering items.

### Disclosures

Several authors of this paper are members of the Arcus collaboration; Should
NASA select Arcus for implementation, their institutions will receive funding
which may be used to fund the authors’ salaries in full or in part in the
future.

### Code, Data, and Materials Availability

Data presented in this paper can be made available by the author (RKH) upon
reasonable request.

### Acknowledgments

This work has been funded in part under NASA grants 80NSSC22K1904 and
80NSSC250K0780 and by the MIT Kavli Insititute for Astrophysics and Space
Research. We appreciate facility support from MIT.nano. This research also
used resources of the Advanced Light Source (beamline 6.3.2), a U.S. DOE
Office of Science User Facility under contract no. DE-AC02-05CH11231.

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Ralf K. Heilmann is a principal research scientist at the MIT Kavli Institute
for Astrophysics and Space Research, and the associate director of the Space
Nanotechnology Laboratory (SNL). He received his Diplom in physics from the
FAU Erlangen/Nürnberg (1991), and his MS (1993) and PhD (1996) in physics from
Carnegie Mellon University. After a postdoc at Harvard he joined the SNL and
has since focused on advanced lithography and the development of x-ray optics
for astronomy. He is a Senior Member of SPIE.

Hans Moritz Günther is a research scientist at MIT. He received his
undergraduate degree (in 2005) and his PhD (in 2009) in physics from the
University of Hamburg, Germany. After that, he worked at the Harvard-
Smithsonian Center for Astrophysics and came to MIT in 2015. He is currently
the lead developer of MARX, the ray-tracing software used for the Chandra
X-ray observatory. His science interests are in star formation using data from
the radio to X-rays.

Mark L. Schattenburg is a senior research scientist at the MIT Kavli Institute
for Astrophysics and Space Research, and director of the Space Nanotechnology
Laboratory (SNL). He received his Ph.D. in physics from MIT (1984). He is a
Fellow of Optica.

Biographies and photographs of the other authors are not available.

###### List of Figures

  1. 1  Arcus Probe schematics. Left: Spacecraft with instruments stowed in launch vehicle fairing. Middle: In space after solar panel deployment deployment. The UV spectrometer is on the left, the XRS on the right. Right: After XRS boom deployment.
  2. 2  Layout of the XRS. Left: Angled view, showing the thermal precollimators and SPO arrays for the four OCs on the left. The grating petals are obscured behind the SPO arrays. An x ray is shown that enters the lower left OC (OC-3) and either lands at the telescope focus (black line to DSA-2) or gets diffracted into a blazed order (rainbow colors to DSA-1). Right: View from the x-ray source into the XRS, showing the four OCs and the two DSAs.
  3. 3  Left: Two different views of an optical channel. The grating petal, holding 40 grating windows with a total of 216 grating facets, follows the shape of the Rowland torus. Middle: Prototype grating window with four $32.5\times 32$ mm2 CAT grating facets. Right: Silicon grating membrane, held by tweezers.
  4. 4  Left: Schematic cross-section through a CAT grating of period $p$. The $m^{\rm{th}}$ diffraction order occurs at an angle $\beta_{m}$ where the path length difference between x rays from neighboring grating slots is $m\lambda$. Shown is the case where $\beta_{m}$ coincides with the direction of specular reflection from the grating bar sidewalls ($|\beta_{m}|=|\alpha|$), i.e., blazing in the $m^{\rm{th}}$ order. Right: Scanning electron micrograph (SEM) of a cleaved CAT grating membrane showing top, cross-section and sidewall views of the 200 nm-period silicon grating bars and their monolithically integrated 5 $\mu$m-period L1 cross supports (x rays enter from the top and leave out the bottom).
  5. 5  Schematic showing the structural hierarchy of a CAT grating membrane (not to scale). See text for details.
  6. 6  Representative diffraction efficiencies of 4 $\mu$m-deep CAT gratings from synchrotron data. Left: Example detector scan for $\lambda=2.5$ nm. DE is strongy blazed for $5^{\rm{th}}$ order at 21%. Summing over orders 4-7 gives a total DE of 32.2%. These values include absorption by L1 bars, but not blockage by the L2 mesh. Right: Comparison of model efficiency (sum of blazed orders collected by XRS readouts), multiplied by Debye-Waller-like roughness factor, for 82% L1 open area fraction with synchrotron spot measurements (sum of same blazed orders as for model) for a $32\times 32$ mm2 CAT grating[26]. The L2 mesh is not considered in this plot.
  7. 7  Spectrum of Al anode from two mutually aligned and simultaneously illuminated CAT gratings, showing the Al-Kα doublet measured in $18^{\mathrm{th}}$ order.[29] Camera pixels are 20 $\mu$m in size. The red line is the natural line shape of the doublet,[30] and the dashed line is the best fit to the data. The black solid line is the curve for the lower $3\sigma$ confidence limit, corresponding to $R_{G}=8157$. The upper $3\sigma$ confidence limit includes $R=\infty$. The dash-dotted lines show the individual K${}_{\alpha_{1}}$ and K${}_{\alpha_{2}}$ components with their natural widths on top of the weak sloped background.
  8. 8  Comparison of TG diffraction efficiencies and figure of merit. Left: For Chandra gratings the sum of $\pm 1^{\mathrm{st}}$ order DEs from calibration files is shown [31]. These DEs include all losses from support structures and gaps between facets. For CAT gratings all orders within the blaze envelope are summed up, and a total loss of 31% from support structures and gaps is assumed. “4 $\mu$m CAT” and “6 $\mu$m CAT” are from synchrotron measurements of CAT gratings with $d=4$ and $\sim 6$ $\mu$m. The latter are from the first prototypes of this thickness.[18, 32] “CAT model” is the sum of blazed order DEs from a model with $d=6$ $\mu$m, $b=40$ nm, and 1.5 nm DW roughness parameter. Right: Figure of merit for weak line detection ($\sim\sqrt{A_{\mathrm{eff}}R}$). The same mirror effective area and constant detector efficiency is assumed for all plots. For CAT gratings, $R=10^{4}$ is assumed as an upper limit, based on conservative analysis of lab measurements [28, 30, 29]. The HETG and LETG instruments complement each other over the shown range. Chandra with CAT gratings would be superior by far for $\lambda>1$ nm due to a combination of higher DE and blazing into higher orders.
  9. 9  Comparison of figures of merit between soft x-ray spectroscopy instruments in operation and the Arcus Probe XRS. “Chandra Gratings” refers to the better of HEG, MEG, and LETG. The curve for the Resolve microcalorimeter on XRISM[34] assumes that the closed aperture door is open. The wavelength ranges for important plasma diagnostics lines are also indicated. Left: Figure of merit for absorption line detection. Right: Figure of merit for emission line detection.
  10. 10  Top down scanning electron micrograph (SEM) images of grating SP3 before (left) and after 20 cycles of HF vapor etching and oxidation in ambient air (right). The 200 nm-period grating bars are slightly thinner after 20 cycles, but the difference is difficult to quantify precisely from SEM images alone.
  11. 11  Diffraction efficiencies at $\lambda=1.75$ nm for orders 0-10 and sum of orders 3-9 as a function of incidence angle. Dashed lines are before thinning, solid lines after thinning. Left: Grating SP3. Right: Grating SP5.