Further author information: (Send correspondence to R.K.H.)
E-mail: ralf at space.mit.edu, , URL: http://snl.mit.edu/

# Toward volume manufacturing of high-performance soft x-ray critical-angle
transmission gratings

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  Jungki Song  Space Nanotechnology Laboratory, MIT Kavli Institute
for Astrophysics and Space Research, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  Matthew T. Cook  MIT Lincoln Laboratory, Lexington,
MA 02421, USA  James A. Gregory  MIT Lincoln Laboratory, Lexington, MA 02421,
USA  Renee D. Lambert  MIT Lincoln Laboratory, Lexington, MA 02421, USA
Dimitri A. Shapiro  MIT Lincoln Laboratory, Lexington, MA 02421, USA  Douglas
J. Young  MIT Lincoln Laboratory, Lexington, MA 02421, USA  Miranda Bradshaw
Max-Planck-Institut für Extraterrestrische Physik, 85748 Garching, Germany
Vadim Burwitz  Max-Planck-Institut für Extraterrestrische Physik, 85748
Garching, Germany  Gisela D. Hartner  Max-Planck-Institut für
Extraterrestrische Physik, 85748 Garching, Germany  Andreas Langmeier  Max-
Planck-Institut für Extraterrestrische Physik, 85748 Garching, Germany
Randall K. Smith  Center for Astrophysics, Harvard-Smithsonian Astrophysical
Observatory, Cambridge, MA 02138, USA  Mark L. Schattenburg  Space
Nanotechnology Laboratory, MIT Kavli Institute for Astrophysics and Space
Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

###### Abstract

High-resolution ($R=\lambda/\Delta\lambda>2000$) x-ray absorption and emission
line spectroscopy in the soft x-ray band is a crucial diagnostic for the
exploration of the properties of ubiquitous warm and hot plasmas and their
dynamics in the cosmic web, galaxy clusters, galaxy halos, intragalactic
space, and star atmospheres. Soft x-ray grating spectroscopy with $R>10{,}000$
has been demonstrated with critical-angle transmission (CAT) gratings. CAT
gratings combine the relaxed alignment and temperature tolerances and low mass
of transmission gratings with high diffraction efficiency blazed in high
orders. They are an enabling technology for the proposed Arcus grating
explorer and were selected for the Lynx design reference mission grating
spectrometer instrument. Both Arcus and Lynx require the manufacture of
hundreds to perhaps $\approx 2000$ large-area CAT gratings. We are developing
new patterning and fabrication process sequences that are conducive to large-
format volume processing on state-of-the-art 200 mm wafer tools. Recent x-ray
tests on 200 nm-period gratings patterned using e-beam-written masks and 4x
projection lithography in conjunction with silicon pore focusing optics
demonstrated $R\approx 10^{4}$ at 1.49 keV. Extending the grating depth from 4
$\mu$m to 6 $\mu$m is predicted to lead to significant improvements in
diffraction efficiency and is part of our current efforts using a combination
of deep reactive-ion etching and wet etching in KOH solution. We describe our
recent progress in grating fabrication and report our latest diffraction
efficiency and modeling results.

###### keywords:

Arcus, Lynx, critical-angle transmission grating, x-ray spectroscopy, blazed
transmission grating, soft x-ray, grating spectrometer, high resolving power

##  1 INTRODUCTION

The soft x-ray band covers the characteristic lines of the most abundant
“metals” (O, C, Ne, Fe, N, Si, Mg, S). Their ionization states and line ratios
provide crucial diagnostics about the plasmas they are part of. High-
resolution absorption and emission spectra tell us about hot gas dynamics at
the edges and farther outside of galaxies, revealing how galaxies have formed
and evolved, and how supermassive black holes at the center of galaxies
interact with their surroundings. They can help us map the distribution of the
missing baryons, presumed to be hiding in the warm-hot intergalactic medium.
On a smaller scale accretion of hot gas during star formation also generates
soft x-ray spectra with rich plasma diagnostics. More detailed descriptions
can be found in work by Smith[1] and Bautz.[2]

Currently operating x-ray spectroscopy missions (the Chandra High Energy
Transmission Grating Spectrometer (HETG)[3] and the XMM-Newton Reflection
Grating Spectrometer (RGS)[4], both launched 20 years ago with technology
developed 30 years ago) lack both the effective area and the resolving power
($R=\lambda/\Delta\lambda$) for these measurements. Luckily, a generation of
technology development later, we can now build an instrument on an Explorer
budget that is much more powerful than what used to require a flagship
mission, as the Arcus[1] soft x-ray grating Explorer mission concept shows.
Arcus features a modular design with four parallel optical channels, each
consisting of a co-aligned array of 12 m-focal length silicon pore optic (SPO)
mirror modules,[5] developed in Europe for the Athena[6] mission. Just aft of
each mirror array follows an array of a few hundred critical-angle
transmission (CAT) gratings arranged on the surface of a tilted Rowland
torus.[7, 8, 9] Arcus will have an effective area of at least 200 cm2 and
$R>2500$ over the 1-3 nm wavelength range, exceeding HETG and RGS figures of
merit by factors of at least 5-10. The soon (2022) to be launched Resolve
microcalorimeter on XRISM[10] will also be far from competitive with Arcus
performance in high-resolution soft x-ray spectroscopy of point sources.

The current Astrophysics Decadal Review is evaluating four Surveyor-class
mission concepts. One of them, the Lynx[11] x-ray observatory, features a
retractable grating array for large-area ($>4000$ cm2) high-resolution
($R>5000$) soft x-ray spectroscopy, covering a fraction of a large x-ray
optic. A CAT grating spectrometer made of $\sim 2000$ large-area gratings[12,
13] was selected for the Design Reference Mission described in the Lynx X-ray
Observatory Report.[14] As part of this effort we developed a detailed
Technology Development Roadmap that can take CAT grating technology from
today’s state-of-the-art to Technology Readiness Level (TRL) 6 for Lynx by
2024.[15]

Technology readiness is a very useful concept regarding the maturation of new
technology. We have previously described CAT grating TRL in more detail.[16]
However, when more than a few manifestations of the technology are needed, the
manufacturability of a large number of very similar or almost identically
performing items needs to be considered as well. As with the mass fabrication
of tens of thousands of mirror segments for a large x-ray mirror,[17] a
manufacturing approach must be developed that can reliably produce many
hundreds to a few thousands of interchangeable grating elements.

Below we first give a brief overview of the CAT grating principle, our
structural design, and our fabrication approach for individual gratings for
technology development. We describe the recently demonstrated addition of
aligned front and back side structures. This is followed by discussion of
process transfer to tools capable of repeatable batch processing of 200 mm
wafers, which will greatly automate and accelerate grating fabrication. We
then present recent measurements of x-ray resolving power of a grating
patterned with the new tool set and discuss the modeling of diffraction
efficiency data before we conclude and summarize.

##  2 CAT GRATING PRINCIPLE AND STRUCTURAL HIERARCHY

CAT gratings are comprised of ultra-high aspect-ratio, freestanding grating
bars with nm-smooth sidewalls. The gratings are rotated by an angle such that
x rays of wavelength $\lambda$ impinge on the sidewalls at graze angles
$\theta$ below the critical angle for total external reflection $\theta_{c}$
(see Fig. 1). The grating equation provides the $m^{\mathrm{th}}$ order
diffraction angle $\beta_{m}$ via

${m\lambda\over p}=\sin\theta-\sin\beta_{m},$ (1)

with $p$ being 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 (typically 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 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. We initially chose a design
with grating period $p=200$ nm, grating bar depth $d=4$ micrometers, and bar
thickness $b\approx 60$ nm. Such a grating is most efficient when
$\tan\theta\approx(p-b)/d$, or $\theta\approx 2$ deg.

---
Figure 1:  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 AA’ and BB’ is $m\lambda$. Shown is the case
where $\beta_{m}$ coincides with the direction of specular reflection from the
grating bar sidewalls ($|\beta_{m}|=|\theta|$), i.e., blazing in the
$m^{\rm{th}}$ order. Right: Scanning electron micrograph 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 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. 1, the bars are held in place by a monolithically integrated
5 $\mu$m-period Level 1 (L1) support mesh. In order to manufacture CAT
gratings that can cover large areas on the order of thousands of square
centimeters, additional support structures are needed for the few-$\mu$m thin
grating layer. Fig. 2 shows an additional, much thicker and stronger Level 2
(L2) hexagonal support structure on the scale of $\sim 1$ mm. The photograph
on the right shows a so-called grating membrane, featuring an additional Level
3 (L3) frame around the edge. This edge is used to bond the membrane to a
metal frame that serves as the mechanical interface to a larger machined
grating array structure (GAS).

---
Figure 2:  Left: Schematic showing the structural hierarchy of a CAT grating
membrane (not to scale). Right: Photograph of an existing 32 x 32 mm2 CAT
grating membrane with back illumination to show the hexagonal L2 mesh, and
with visible light diffraction due to the L1 mesh.

##  3 GRATING FABRICATION APPROACH

We have fabricated numerous CAT gratings in sizes up to $32\times 32$ mm2 in
our lab and other MIT campus facilities, using 100 mm-diameter silicon-on-
insulator (SOI) wafers with a (110) device layer (front side) of thickness
$d$, a thermal oxide layer on both sides, and an additional plasma-enhanced
chemical vapor deposition (PECVD) oxide layer on the handle layer (back side)
(see Fig. 3).[18, 19] The grating pattern is defined with interference
lithography. The grating lines are aligned parallel to one set of vertical
{111} planes. The L1 cross-support mesh is defined in similar fashion. The
combined pattern is transferred into the thermal oxide, which serves as a mask
for the subsequent deep reactive-ion etch (DRIE). The buried oxide (BOX) layer
serves as an etch stop. A short anisotropic wet etch in KOH solution greatly
reduces the scalloping from the DRIE and leaves much smoother {111} grating
bar sidewalls. The front side is protected and bonded to a carrier wafer. The
L2 hexagon mesh is patterned into the back side PECVD oxide, surrounded by a
narrow frame area. This pattern is etched through the $\approx$ 500
$\mu$m-thick handle layer using DRIE, again using the BOX layer as an etch
stop. Then the wafer is detached from the carrier, cleaned, and critical-point
dried. The BOX layer is removed from the open areas using RIE and HF vapor
etching, resulting in a freestanding grating with integrated L1 cross-supports
and a strong L2 mesh that keeps large-area gratings supported (see Fig. 2).

---
Figure 3:  Highly simplified schematic overview of CAT grating fabrication
steps. See text for more details.

##  4 aligned front side L2 mesh

At the end of the above fabrication sequence device layer and handle layer are
only connected by buried oxide in the areas where the front side L1 mesh and
the back side L2 mesh and L3 frames overlap (see red arrow in step B2 of Fig.
3). During the isotropic HF vapor etch, undercutting can lead to thinning of
the connected areas or even detachment between device and handle layers. It is
difficult to time the HF etch perfectly right, and variations in grating bar
widths and device layer and BOX thickness may lead to leftover oxide between
grating bars in some areas and detachment in others. A better approach for
higher vapor HF process latitude and improved mechanical behavior is to
incorporate an L2 mesh in the front side pattern that has the same periodicity
and is aligned with the back side L2 mesh. This requires a third front side
lithography sequence.

In order to incorporate a L2 mesh on the front side, the back side pattern
included a set of alignment marks near the edge of the wafer. An additional
layer of photoresist was applied to the front side after the CAT and L1 lines
were patterned into the front-side thermal oxide layer. This photoresist was
patterned with a L2 and L3 mesh and aligned to the back side via the alignment
marks. The front side was then etched via the standard DRIE and subsequent
processing steps depicted in Fig. 3, starting on step F3. The resulting
grating was significantly easier to process and the vapor HF step did not
damage the membrane via the undercutting observed without the front-side L2
mesh. Future gratings patterned via optical-projection lithography will have
the front-side L2 mesh in the same mask as the CAT and L1 support, enabling a
much simpler process than the process demonstrated via interference
lithography and multiple mask layers.

---
Figure 4:  CAT grating membrane ($\sim 10\times 30$ mm2) with aligned front
(left image) and back side (right image) L2 mesh.

##  5 Process transfer to 200 mm state-of-the-art tools

---
Figure 5:  Left: 200 mm wafer front side oxide mask patterned at LL with CAT
grating and L1 mesh. The rainbow pattern is due to diffraction from the L1
mesh. Wagon wheel patterns are visible around the perimeter. Right: 200 mm
wafer with back side L2 and L3 pattern in resist written by an MLA tool in
MIT.nano.

For a mission such as Arcus or Lynx, on the order of $\sim 700-2000$ grating
facets have to be manufactured with reliable and repeatable performance over a
limited period of time. In contrast, most of the gratings we have fabricated
so far were made in a manual and time-consuming “beaker and tweezer” fashion,
often using shared facilities and tools with limited repeatability. Scaling up
CAT grating fabrication requires a more efficient approach with more
repeatable performance, more predictable outcomes, and lower costs. For this
reason we have begun a collaboration with the MIT Lincoln Labs (LL) Advanced
Imager Technology Group and the LL Microelectronics Laboratory. This group has
automated coating, patterning and etching tools for the batch processing of
200 mm wafers. At the same time the new MIT.nano facility on the MIT campus is
bringing new 200 mm tools online that can be used for post-LL process steps.

A key to achieving repeatable fabrication outcomes for the complex CAT grating
fabrication sequence is the initial patterning step. The grating period has to
be repeatable over all lithographic exposures and constant over all areas as a
necessary requirement for achieving high resolving power. The grating etch
masks must have a well-defined duty cycle to enable reliable outcomes for the
challenging device layer DRIE step. The LL ME group has a 193 nm 4X projection
lithography stepper/scanner tool for 200 mm wafers. We performed initial tests
on this tool using an electron-beam (e-beam) written mask with integrated CAT
grating and L1 patterns. The tool exposure field can be stepped over the whole
wafer, placing a nearly continuous and constant pattern almost everywhere.
Instead of one Arcus-size grating from the center of a 100 mm wafer using
interference lithography we hope to obtain 16-20 gratings out of a single 200
mm wafer (see Fig. 5).

LL has developed a layer stack of amorphous Si and silicon nitride to transfer
the photomask pattern from the photoresist into the 300 nm thick thermal oxide
layer. X-ray diffraction from the device layer has been used to measure the
angle between the {111} planes and the wafer notch and to clock the wafer
accordingly for the CAT grating line pattern to be parallel to the {111}
planes. The photomask also contains spoked wagon wheel patterns[18] that have
been used to verify proper clocking.

The first few wafers patterned with this mask have mainly been used for front
side DRIE development, using both four and six $\mu$m thick device layers. The
mask has successfully been used to etch CAT grating bars to a depth of six
$\mu$m over full 200 mm SOI wafers in a state-of-the-art DRIE tool located
abroad. Preliminary work on KOH polishing has been started as well as efforts
to measure the grating bar tilt angles[20] along full 200 mm wafers.

Our latest photomask contains an L2 mesh (with 40 $\mu$m wide hexagon beams)
in addition to the CAT grating and L1 mesh patterns (see Fig. 6). We have
demonstrated front side patterning across a whole 200 mm wafer with phase
continuity of all three patterns in both directions. Each exposure field has a
solid 5 $\mu$m boundary along its edges. The resulting crosses at the corners
of four adjacent fields can be used as alignment marks for the back side
patterning of the L2 mesh (with beams wider than 40 $\mu$m). The L2 and L3
patterns are to be written by laser on the back side using a maskless aligner
(MLA) tool in the MIT.nano facility. The facility is new and many of the tools
are 200 mm capable. The PECVD oxide labeled in Fig. 3 has also been etched on
full 200 mm wafers via a SAMCO inductively coupled plasma etch tool.

---
Figure 6:  Left: Result of photoresist exposure dose study across a 200 mm
wafer with resist/ARC/Si3N4/a-Si/thermal oxide stack, using the CAT/L1/L2
front side mask. Middle: Microscope image of the patterned resist at the
intersection of four exposure fields, showing phase continuity of the ($\sim
1$ mm-pitch) L2 mesh and the solid boundaries along the field edges. Right:
Top down scanning electron micrograph, showing CAT grating and L1 and L2
support structure patterns in resist near an inside corner of the L2 hexagon
mesh. (The thin dark horizontal line is an image artifact.)

##  6 CAT grating resolving power

An x-ray grating spectrometer typically consists of focusing optics, an
objective grating array just downstream of the optics, and a readout camera
that records the dispersed source spectrum. Assuming perfect gratings, the
resolving power is to first order simply given by the diffraction angle of the
order of interest, divided by the FWHM of the angular PSF of the focusing
optics in the dispersion direction. However, grating imperfections can lead to
additional broadening of the spectral features, thereby reducing the resolving
power. Often this broadening is modeled as a variation of the grating period
$\Delta p$. Previously we demonstrated $R>10{,}000$ for a $\sim 10\times 30$
mm2 CAT grating that had been patterned by interference lithography and
illuminated by a single slumped glass mirror pair over an area of $\sim
1\times 30$ mm2.[21, 22]

Interference lithography provides very smooth and well-understood phase
progression over reasonably large areas, but can suffer from hyperbolic
distortions, depending on the details of the setup and the desired grating
size. In the semiconductor industry patterning is primarily done using
elaborate stepper/scanner tools with nm pattern placement precision over a
roughly $26\times 32$ mm2 exposure field, for example. The pattern itself is
defined in a mask that is written at four times the scale of the desired
patterns, using electron beam (or e-beam) lithography. Ideally, for a grating
patterned in this fashion, local and global deviations $\Delta p$ from the
desired grating period $p$ should be smaller than $p/R$ over the whole
exposure field. In practice it is sufficient for the standard deviation
$\sigma_{p}$ of the period distribution to satisfy $\sigma_{p}<p/R$, which is
20 pm in our case.

The most convincing way to test the grating performance is to measure the
broadening due to $\sigma_{p}$ in an x-ray grating spectrometer. We therefore
performed a measurement very similar to our previous work,[22] with some
important differences. We did not have any 200 mm SOI wafers with a $<$110$>$
device layer available at the time. Instead we used one of two $<$100$>$ 200
mm SOI wafers with a nominally 3 micron thick device layer. We patterned it
using the LL 4X optical projection lithography stepper/scanner using an e-beam
written mask with a 50% duty cycle CAT grating pattern and an orthogonal 15%
duty cycle L1 support mesh pattern. After projection this results in a 200 nm-
period grating with a 5 $\mu$m-period orthogonal support mesh in resist. This
pattern was transferred into an underlying oxide layer, which then served as
the hard mask for the DRIE through the device layer, stopping on the BOX. Wet
etching in KOH solution was not performed, since the device layer had the
wrong crystal orientation. A hexagonal L2 support mesh and an L3 frame were
etched into the 625 $\mu$m thick handle layer, resulting in a silicon grating
membrane with $\sim 26\times 27$ mm2 of area inside the frame. The BOX was
removed from the open areas using RIE and a vapor HF etch. The resulting
silicon grating membrane is shown on the left of Fig. 7. The grating was then
epoxied to a Ti frame at a single point in the center of one of the L3 edges.

The x-ray test was performed at the PANTER facility in Neuried, Germany (see
middle of Fig. 7). As a focusing optic we used a 12 m-focal length SPO x-ray
optical unit (XOU) with 14 stack pairs, which provided a roughly $11\times
26.2$ mm2 illumination area for the grating, and a best focus of 1.04 arcsec
(FWHM) along the dispersion direction. The grating was placed $\sim 268$ mm
from the XOU node, which in turn was about 119.5 m from the electron-
bombardment x-ray source. We used an Al target, which emitted the
characteristic K$\alpha_{1,2}$ doublet at 1.49 keV. As in our previous
studies, we focused our effort on the 18th diffraction order at 4.3 degrees
from the straight-through 0th order. The grating was rotated by half this
angle for optimal blazing. Nevertheless, the diffraction efficiency in such a
high order was rather low due to two main reasons: This angle exceeds the
critical angle for Si at this wavelength by a factor of $\sim 2$, and the
grating bar sidewalls were rough from the DRIE.

The data shown on the right in Fig. 7 was collected using the PIXI detector
(PI-MTE 1300B, $1340\times 1300$ 20 $\mu$m square pixels, placed 13345 mm from
the mirror node due to the finite source distance) with a total exposure time
of just over seven hours, with data taken on two different days simply summed
up. The data was projected onto the dispersion axis and fitted to a model of
the K$\alpha_{1,2}$ doublet with parameters from Ref. 22 that is convolved
with a Gaussian of variable width, representing broadening due to all sources
(source size, optics PSF, grating period variations, aberrations, drifts in
the setup, etc.). The only other fit parameters are the position of the
K$\alpha_{1}$ peak and an overall scale factor. After fitting we subtract out
(in rss fashion) the measured width of the direct beam, assuming Gaussian
shapes for the beam and all other sources of broadening. We find the best-fit
standard deviation of the remaining Gaussian to be $\sim 52$ $\mu$m on the
detector. If we interpret this broadening as simply due to grating period
variations we find that this grating is compatible with a resolving power of
$R\geq 9470$. When we analyze the data collected on two different days
separately we find $R\geq 9800$ and $R\geq 12000$, with a $\sim 10$ $\mu$m
shift in best fit peak position between the two days. At this point in our
analysis it is unclear how much of this shift is statistical variation and to
what degree it may present small dimensional changes in the setup over tens of
hours in time.

This result clearly shows that gratings patterned with an e-beam written mask
and 4X projection lithography are suitable for high-resolution spectrographs
requiring $R>7500$. The grating was illuminated with 14 confocal mirror pairs,
and the sampled area was $\sim 10$ times larger than in our previous work. We
don’t see any fundamental problems that would prevent this technology from
being scaled to the size required for Arcus or Lynx.

---
Figure 7:  Left: CAT grating membrane fabricated from a 200 mm SOI wafer with
3 $\mu$m device layer. The outer dimensions are $\sim 30.6\times 29.5$ mm2.
The blue rectangle shows the approximate area illuminated by x rays. Holes are
mainly due to damage from manual handling. Middle: View towards the source at
PANTER during laser alignment. The grating under test is in the lower left
quadrant of the grating plate. The upstream SPO XOU is visible through the
open upper left quadrant. The red dot in the background is an alignment laser
emanating from the source end of the facility and illuminating the XOU. Red
laser light transmitted through the XOU can be seen hitting the grating.
Right: Measured K$\alpha$ spectrum in 18th order projected onto the dispersion
axis versus detector pixel number. The data is shown with a dash-dotted line,
the dashed line shows the theoretical spectrum using natural line widths, and
the solid line is the fit to the data, clearly showing the shoulder in the
doublet.

##  7 CAT grating diffraction efficiency: data and modeling

In order to design CAT gratings for different applications it is imperative to
have a good model that can predict diffraction efficiency as a function of
structural and geometric parameters such as grating and coating materials,
grating period, bar width and depth, and x-ray wavelength and angle of
incidence. We have used rigorous coupled-wave analysis (RCWA) for most of our
modeling. The grating structure is modeled as a sequence of stratified layers,
with distinct regions of different index of refraction within each grating
layer. Periodic boundary conditions are assumed in the direction of the
grating vector. Thus grating bars are modeled with a rectangular profile (for
example, width of 60 nm, depth of 4 $\mu$m) in the case of a single grating
layer. Other bar profiles can be approximated as “pyramids” of rectangles
comprised of many layers.[23, 24]

Most of our diffraction efficiency measurements to date have been performed at
the calibration and standards beamline 6.3.2 at the Advanced Light Source at
Lawrence Berkeley National Laboratory. Many measurements have been published
previously, typically in the range of 80-100% of predictions for perfect
structures, but often lower closer to the critical angle or critical
wavelength.[21, 25, 26, 27, 28, 29] We typically sample gratings with a $\sim
50\times 250$ $\mu$m2 beam size in areas between the L2 supports. A footprint
of this size averages over many tens of L1 support bars and therefore properly
reflects their impact on grating effective area. For mission purposes and
effective area predictions we also need to take blockage by L2 structures into
account. This could simply be calculated from the lithographically defined
dimensions of the L2 mask. However, pattern transfer and deep etch
imperfections could lead to deviations from the theoretically expected
structure. In a recent experiment we therefore stepped gratings across the
x-ray beam in many small (50-200 $\mu$m) steps over areas containing $\sim 20$
L2 hexagons and measured the diffraction efficiency for several orders at each
spot at a few angles of incidence. The left of Fig. 8 shows the sum of
absolute efficiencies of orders 3-8 as a function of position on the grating
for the angle of incidence with the highest efficiency. Clearly the blockage
from the hexagonal L2 mesh can be seen as a hexagonal pattern of reduced
efficiency. Nevertheless, the average absolute efficiency over this $\sim
4\times 4$ mm2 area is 31.6% (including L1 and L2 blockage), which already
exceeds the Arcus goal of 28%. If we divide the average efficiency over some
area by the highest efficiency found over the same area we obtain values in
the range of 74-79%, which is consistent with the L2 design open area of 81%.

---
Figure 8:  Synchrotron data from grating X14 at an x-ray wavelength of 2.38
nm. Left: Sum of absolute efficiencies of orders 3-8 as a function of beam
footprint position (in mm) on the grating at an incidence angle of 1.7
degrees, including L1 and L2 blockage. Clearly, the impact of the hexagonal L2
support mesh can be seen on the mm-scale. Right: Absolute diffraction
efficiency (including L1 blockage) for orders zero through nine on a single
location on the grating as a function of incidence angle (“rocking scan”).
Dashed lines are measured efficiencies and solid lines are model efficiencies.
See text for details.

The grating bar profile cannot be discerned in sufficient detail from SEM
images alone to predict diffraction efficiency with better than $\sim 10$%
relative accuracy. We are investigating how well we can model fabricated CAT
grating bar profiles from rocking scans of zeroth order at a single x-ray
wavelength of interest and predict diffraction efficiency for higher orders.
The right of Fig. 8 shows example data (dashed lines) from grating X14 where
we use a three-layer fit to the zeroth order rocking scan to find grating bar
values for each layer (layer depth and grating bar width). The effect of L1
supports is modeled with a single number, representing the percentage of
grating area occupied by the L1 mesh (L1 duty cycle). The L2 mesh is not
considered, since this data was taken near the center of an L2 hexagon. The
result of the zeroth order fit is then used to model the efficiency of the
higher diffraction orders. RCWA assumes perfectly rectangular profiles for
each grating layer and predicts higher efficiencies than the measured ones. We
model grating bar imperfections such as roughness heuristically as a single
wavelength- and angle-dependent Debye-Waller factor $f_{\mathrm{DW}}$, which
has no impact on zeroth order, but decreases the efficiency of higher orders
according to

$f_{\mathrm{DW}}=\exp{(-q_{z}^{2}\sigma^{2})}=\exp{\left[-\left({2\pi
m\sigma\over{p}}\right)^{2}\right]},$ (2)

where $q_{z}=q_{z}^{out}-q_{z}^{in}=(2\pi/\lambda)(\sin\beta_{m}-\sin\theta)$
is the wavevector transfer parallel to the grating vector and $\sigma$ is a
roughness parameter. The solid lines on the right of Fig. 8 were generated
using $\sigma=2.6$ nm, which is a reasonable number if we chose to interpret
it as rms sidewall roughness. The total depth of this grating is found to be
4.0 $\mu$m, and the grating bar widths range from 43 nm at the side facing the
x-ray source to 38 nm on the side facing the detector. The L1 duty cycle is
found to be 16%. The agreement between data and model is quite good over many
orders and a wide range of incidence angles, giving us confidence in our
modeling approach.

##  8 Discussion and outlook

CAT grating technology has progressed significantly over the last decade. We
believe its TRL is close to six for the purposes of the Arcus mission. Arrays
of up to four aligned gratings have been built and shown to perform within
expectations.[29, 7] Our fabrication approach is relatively mature, but we
continue to look for improvements that lead to shorter fabrication times and
better and more reliable outcomes. As an example, the addition of a front side
L2 mesh, aligned with the back side L2 mesh, leads to wider process latitude
and higher mechanical strength. Once added to the photomask pattern it
requires no additional fabrication steps (besides alignment of the back side
L2 mesh).

Manufacturing hundreds or thousands of CAT gratings demands a more advanced
tool set, larger wafers, and a larger degree of automation and repeatability.
Transferring a complex multi-step nanofabrication process to a different tool
set is not trivial. The demands for process uniformity across 200 mm wafers
are challenging, even for state-of-the-art tools. Nevertheless, we have made
steady progress in moving from 100 mm wafers processed on research tools open
to the whole campus community to processing 200 mm wafers on more restricted
and better controlled production style tools. Optical 4X projection
lithography with commercially e-beam written masks has produced grating areas
of significant size with resolving power on the order of $10^{4}$. To the best
of our knowledge such x-ray grating resolving power has never been shown
before in combination with e-beam patterning.

The use of a mask with a fixed, precise angle between CAT grating bars and L1
mesh bars opens up the opportunity for high-precision roll alignment between
gratings using visible-light diffraction from the L1 mesh. This can be done in
a normal incidence transmission geometry, which is much less sensitive to
grating non-flatness than methods based on reflection.[30] We are currently
exploring this alignment method.

The back side pattern defines the back side L2 mesh, which is aligned to the
front side L2 mesh, which in turn has a fixed orientation with respect to the
CAT grating bars. It also defines the L3 frame of a grating membrane with
lithographic precision. This means we can fabricate macroscopic precision
structures (the outside dimensions of the L3 frame) with a precisely defined
orientation relative to the nm-scale CAT grating bars. In principle our
fabrication scheme makes it possible to use the L3 edges as reference surfaces
for fast and precise grating alignment and integration into large arrays.

We expect e-beam patterning of masks for chirped gratings to be feasible as
well. The ability to chirp would allow us to make larger (and thus fewer)
gratings for Lynx, without introducing unacceptable losses in $R$ caused by
aberrations due to the deviation of flat gratings from the Rowland torus
surface.[12, 31]

We continue to work on transferring more of the fabrication steps onto 200 mm
tools for faster and cheaper production. Process uniformity over the large
area of a 200 mm wafer remains a challenge and may limit the number of quality
gratings that can be extracted from a single wafer.

Individual CAT grating performance exceeds currently envisioned resolving
power requirements for both Arcus and Lynx. Diffraction efficiency meets Arcus
requirements. Lynx demands higher diffraction efficiency, which can be
achieved by moving from 4 $\mu$m to 6 $\mu$m grating depth. We are well on our
way toward this goal. CAT grating DRIE down to 6 $\mu$m depth on SOIs has been
accomplished, and we have begun KOH polishing on those gratings. In addition,
for Lynx the fraction of grating area blocked by L1 and L2 structures also
needs to be reduced from 15-20% to the 10% level per mesh while still
providing enough mechanical strength. As part of this work we have performed
detailed investigations of alternative L2 structures using analytical and
finite-element approaches.[32]

We are progressing toward a situation where we can produce a significant
number of CAT gratings with very similar structural parameters, which will
allow us to perform more systematic experiments and process development.
Hopefully this will also provide us with a sufficient number of gratings to
test the performance of progressively larger grating arrays that can serve as
believable surrogates for fully sized proposed instruments.

###### Acknowledgements.

We gratefully acknowledge facility support from Microsystems Technology Labs,
the Nanostructures Lab and MIT.nano, all at MIT, and SAMCO, Inc. We thank Eric
Gullikson for support at the ALS. A part of this work used resources of the
Advanced Light Source, which is a DOE Office of Science User Facility under
contract no. DE-AC02-05CH11231. Part of the work performed at PANTER has been
supported by the European Unions Horizon 2020 Programme under the AHEAD
project (grant agreement n. 654215). This work was supported by NASA grants
NNX17AG43G, 80NSSC19K0335, and 80NSSC20K0780.

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