figure table

# A Comprehensive Line-Spread Function Error Budget for the Off-Plane Grating
Rocket Experiment

Benjamin D. Donovan  The Pennsylvania State University, University Park, PA
16802  Randall L. McEntaffer  The Pennsylvania State University, University
Park, PA 16802  James H. Tutt  The Pennsylvania State University, University
Park, PA 16802  Bridget C. O’Meara  The Pennsylvania State University,
University Park, PA 16802  Fabien Grisé  The Pennsylvania State University,
University Park, PA 16802  William W. Zhang  NASA Goddard Space Flight Center,
Greenbelt, MD 20771  Michael P. Biskach  NASA Goddard Space Flight Center,
Greenbelt, MD 20771  Timo T. Saha  NASA Goddard Space Flight Center,
Greenbelt, MD 20771  Andrew D. Holland  The Open University, Walton Hall,
Milton Keynes, UK  Daniel Evan  The Open University, Walton Hall, Milton
Keynes, UK  Matthew R. Lewis  The Open University, Walton Hall, Milton Keynes,
UK  Matthew R. Soman  The Open University, Walton Hall, Milton Keynes, UK
Karen Holland  XCAM Ltd., Northampton, UK  David Colebrook  XCAM Ltd.,
Northampton, UK  Fraser Cooper  XCAM Ltd., Northampton, UK  David Farn  XCAM
Ltd., Northampton, UK

###### Abstract

The Off-plane Grating Rocket Experiment (OGRE) is a soft X-ray grating
spectrometer to be flown on a suborbital rocket. The payload is designed to
obtain the highest-resolution soft X-ray spectrum of Capella to date with a
resolution goal of R($\lambda/\Delta\lambda$) $>2000$ at select wavelengths in
its 10–55 $\mathrm{\SIUnitSymbolAngstrom}$ bandpass of interest. The optical
design of the spectrometer realizes a theoretical maximum resolution of
$R\approx 5000$, but this performance does not consider the finite performance
of the individual spectrometer components, misalignments between components,
and in-flight pointing errors. These errors all degrade the performance of the
spectrometer from its theoretical maximum. A comprehensive line-spread
function (LSF) error budget has been constructed for the OGRE spectrometer to
identify contributions to the LSF, to determine how each of these affects the
LSF, and to inform performance requirements and alignment tolerances for the
spectrometer. In this document, the comprehensive LSF error budget for the
OGRE spectrometer is presented, the resulting errors are validated via
raytrace simulations, and the implications of these results are discussed.

###### keywords:

error budget, X-ray spectroscopy, suborbital rocket, reflection gratings,
mono-crystalline silicon X-ray optics, electron-multiplying CCDs

*Benjamin D. Donovan<EMAIL_ADDRESS>
##  1 Introduction

The Off-plane Grating Rocket Experiment (OGRE) is a soft X-ray grating
spectrometer that will be flown on a suborbital rocket. With a spectral
resolution requirement of $R(\lambda/\Delta\lambda)>1500$ across its 10 – 55
$\mathrm{\SIUnitSymbolAngstrom}$ bandpass of interest and a goal of $R>2000$
at select wavelengths in this same bandpass, OGRE will obtain the highest-
resolution soft X-ray spectrum of Capella to date. This performance will
enable OGRE to examine the spectrum of its target, Capella ($\alpha$ Auriga),
in unprecedented detail. This detailed observation will permit existing line
blends in the soft X-ray spectrum of Capella to be resolved, new and updated
emission lines to be integrated into plasma spectral models, and more accurate
plasma characteristics to be determined for this source [1].

To achieve its performance goal of $R>2000$, OGRE will utilize three cutting-
edge technologies: a mono-crystalline silicon X-ray optic assembly
manufactured by NASA Goddard Space Flight Center (GSFC)[2], six reflection
grating modules developed by The Pennsylvania State University operated in the
extreme off-plane mount [3] and integrated into a grating assembly, and an
array of four electron-multiplying CCDs (EM-CCDs) manufactured by e2v and
integrated into a detector assembly by XCAM Ltd. and The Open University [4].

###  1.1 Optical Design of the OGRE Spectrometer

The OGRE spectrometer employs the traditional three-component X-ray grating
spectrometer design consisting of an X-ray optic that focuses the incident
light from Capella, a reflection grating array that diffracts this light into
its component spectrum, and an array of detectors at the focal plane to sample
the spectrum.

The X-ray optic on board the OGRE spectrometer will be a mono-crystalline
silicon X-ray optic assembly developed by NASA GSFC [2]. The optic assembly
will consist of twelve Wolter I-type[5] (paraboloid + hyperboloid) mirror
shells with radii from $r_{0}=162.47-191.364$ mm and a common focal length of
$Z_{0}=3500$ mm. Each of these shells will be constructed from individual
silicon mirror segments that span 30$\mathrm{\SIUnitSymbolDegree}$ in azimuth.
A more detailed description of this optic assembly can be found in Donovan et
al. (2019) [1].

The OGRE spectrometer will utilize reflection gratings operated in the extreme
off-plane mount to disperse the converging light from the OGRE optic assembly
into its component spectrum. Gratings in the extreme off-plane mount are
oriented quasi-parallel to the groove direction and at grazing incidence
relative to the incident X-ray photons as depicted in Figure 1. Diffraction
then follows the generalized grating equation:

Figure 1: The diffraction geometry of a reflection grating in the extreme off-
plane mount [6, 7]. Light is incident from point A onto the grating surface at
a graze angle $\eta$ and relative to the groove direction by an angle $\Psi$.
Equivalently, this incidence geometry can be described in the spherical
coordinate system by azimuth angle $\alpha$ and polar angle $\gamma$.
Diffraction follows the generalized grating equation (Eq. 1) and light is
diffracted a distance $L$ to azimuth angle $\beta$ on the focal plane. The
dispersion distance between the $n=0$ reflection and $n$-th diffraction order
on this focal plane is given by $x=n\lambda L/d$.

$\sin\alpha+\sin\beta=\frac{n\lambda}{d\sin\gamma},$ (1)

where $\alpha$ is the incident azimuthal angle, $\beta$ is the diffracted
azimuthal angle, $\gamma$ is the polar angle between the incident light and
the groove direction, $d$ is the groove period, $n$ is the diffraction order,
and $\lambda$ is the wavelength of the light [8]. By differentiating Eq. 1
with respect to $\lambda$, it can be shown that:

$\frac{d\lambda}{dx}=\frac{10^{7}}{nLD}\frac{$\mathrm{\SIUnitSymbolAngstrom}$}{$\mathrm{mm}$},$
(2)

where $L$ is the distance a photon on the grating travels to the spectrometer
focal plane, $D$ is the groove density ($\equiv 1/d$), and $x$ is the distance
a photon is diffracted from the $n=0$ reflection
($=L\sin\gamma(\sin\beta+\sin\alpha)$). This equation shows that while the
extreme off-plane mount diffracts its incident light conically, the spectral
information is contained only in one dimension – the dispersion direction ($x$
in Figure 1).

In the OGRE spectrometer, the full 360$\mathrm{\SIUnitSymbolDegree}$ azimuthal
span of the optic assembly is divided into 60$\mathrm{\SIUnitSymbolDegree}$
azimuthal sections. Behind each of these azimuthal sections is a OGRE grating
module containing 60 individual reflection gratings organized into two side-
by-side grating stacks. The grating positions in each module are numerically
optimized to realize maximum spectral resolution at the Fe XVII emission line
($\lambda=15.01$ $\mathrm{\SIUnitSymbolAngstrom}$) – the brightest line
expected to be observed from Capella. Light from two neighboring grating
modules is diffracted to the same location on the focal plane where the
spectra are read out by a single detector. A schematic of this diffraction
geometry can be seen in Figure 2. Spectral isolation on this detector will be
attained through a combination of the detector’s energy resolution and a
slight offset ($\sim 2-3$ mm) of the diffraction arcs relative to one another.
This geometry is repeated two additional times around the optic assembly for a
total of six grating modules diffracting to three separate spectral detectors
(also shown in Figure 2). Further discussion of this geometry can be found in
Donovan et al. (2019) [1].

Figure 2: Diffraction geometry for neighboring grating modules in the OGRE
soft X-ray spectrometer [1]. The optic assembly is divided into six
60$\mathrm{\SIUnitSymbolDegree}$ azimuthal sections. Behind each of these
sections is a grating module containing 60 gratings. To maximize diffraction
efficiency, each grating module operates in the Littrow mount which requires
$\alpha=\beta=\delta$. For the OGRE spectrometer,
$\alpha=\beta=\delta=30$$\mathrm{\SIUnitSymbolDegree}$ such that the
neighboring grating modules diffract to the same location on the focal plane.
This geometry is repeated two times to populate the entire
360$\mathrm{\SIUnitSymbolDegree}$ azimuthal span of optic with grating
modules. Depicted in the bottom-right is the coordinate system for this
schematic view with +ẑ pointing into the page.

The optical design of the OGRE spectrometer realizes a maximum spectral
resolution of $R\approx 5000$ at the Fe XVII emission line. This resolution is
only attained though if all spectrometer components perform flawlessly, if
these components are all perfectly aligned into the spectrometer, and if the
payload remains oriented exactly towards its target. In reality, however, the
measured performance of the gratings in the OGRE grating modules and the
mirror segments in the OGRE optic assembly will deviate from their idealized
performance. These performance errors will begin to degrade the diffracted
line-spread function (LSF) of the spectrometer as observed on the focal plane.
Furthermore, the spectrometer components will not be aligned perfectly to one
other. These misalignments will further decrease the achievable performance of
the spectrometer. Finally, the spectrometer will not remain perfectly pointed
at Capella during its observation, but will instead dither about its ideal
pointing. This in-flight jitter will further degrade the observed LSF. These
three sources of error will all conspire to cause the spectrometer to not
achieve its theoretical spectral resolution of $R\approx 5000$, but to achieve
a performance below this level.

To understand how each misalignment and error contributes to the performance
of the OGRE spectrometer, a comprehensive LSF error budget is required. This
error budget identifies each contribution to the LSF, determines how each of
these contributions affects the observed LSF, and then ultimately assigns
requirements for each potential error or misalignment in the OGRE spectrometer
so that the spectrometer can meet its overall performance requirement of
$R>1500$. In this manuscript, the comprehensive LSF error budget for OGRE is
described and implications resulting from this error budget are discussed.

##  2 The Comprehensive LSF Error Budget

The comprehensive LSF error budget considers potential misalignments and
performance errors in the spectrometer. These misalignments and performance
errors come from each of the three main components of the spectrometer: the
OGRE optic assembly, the OGRE grating modules, and the OGRE detector assembly.
An additional error arises from the in-flight pointing error of the payload.
Each of these misalignments and performance errors modifies the observed LSF
from the idealized LSF by increasing its extent in the dispersion and/or
cross-dispersion direction. This behavior directly impacts the achievable
spectral resolution and/or the effective area of the spectrometer.
Additionally, misalignments can shift the centroid of the LSF on the detector.
These shifts can move important spectral lines of Capella off of the detector,
impacting the science return of the spectrometer. Thus, each potential
performance error and misalignment must be analyzed to ensure the OGRE
performance requirements are met.

Figure 3: Results from a raytrace simulation of the OGRE spectrometer showing
the effect of a yaw misalignment of individual gratings in the grating module
on the extent of the OGRE LSF in the dispersion (blue) and cross-dispersion
(orange) directions. The contribution in the dispersion direction is measured
as a full-width at half-maximum (FWHM), while the contribution in the cross-
dispersion direction is measured as a half-power diameter (HPD). Here, it can
been seen that a grating-level yaw misalignment at these simulated
misalignment values does not contribute to the LSF extent in the dispersion
direction, but it has a noticeable impact on the LSF extent in the cross-
dispersion direction.

The construction of the LSF error budget begins with the consideration of a
single contribution. For example, consider the yaw misalignment (rotation
about ŷ; degrees of freedom defined in Figure 2) of the 60 gratings within the
two grating stacks that form Grating Module #1 in Figure 2. Each of these
gratings has a nominal yaw orientation within its stack, but each grating can
be misaligned relative to this nominal orientation. To simulate the effect of
this misalignment on the final LSF of the spectrometer, a custom raytrace
modelaaaBased upon PyXFocus: https://github.com/rallured/PyXFocus of the OGRE
spectrometer is utilized. A range of potential yaw misalignment values are
simulated in this spectrometer model and the resulting LSF is measured for
each. The results of these simulations are shown in Figure 3. In this figure,
it can be seen that a yaw misalignment of the gratings within the two grating
stacks affects the extent of the OGRE LSF in the cross-dispersion direction,
but has no impact on the extent of the LSF in the dispersion direction. This
agrees with what is expected from theory. Based on these results, a yaw
alignment tolerance for the gratings within a stack would then be set (to be
discussed in Section 2.2.3). Similar simulations would then be run for the
remaining five degrees of freedom (x̂, ŷ, ẑ, pitch, and roll) of the gratings
within their stacks, and requirements for these misalignments would be derived
as well. Further simulations would then be performed for all remaining
misalignments and performance errors within the spectrometer.

Throughout the construction of the error budget, the performance requirements
and goals are continually referenced to ensure that the spectrometer will meet
these requirements and goals. For OGRE, the spectral resolution goal is
$R>2000$ (resolution requirement: $R>1500$) which limits the extent of the LSF
in the dispersion direction. In the cross-dispersion direction, the entire LSF
must remain within the planned window size of the spectral detector ($\sim 6$
mm). While the LSF must remain within this window, movement of the LSF
centroid in the cross-dispersion direction can be larger than this since the
window can be moved on the detector. Total movement is limited to $\sim\pm 12$
mm since the LSF must still remain on the detector ($\sim 25$ mm x $25$ mm;
e2v CCD207-40[4]).

In the following subsections, individual contributions to the OGRE LSF error
budget will be discussed. The discussion here is limited to a single
60$\mathrm{\SIUnitSymbolDegree}$ azimuthal section of the OGRE optic assembly,
a single OGRE grating module placed behind this optic section, and a single
spectral detector – an OGRE spectrometer “channel”. While this analysis is
limited to a single channel, each channel is identical or mirrored with
respect to the modeled channel, so the described analysis will apply to the
remaining five spectrometer channels as well. Furthermore, this analysis is
limited to the wavelength dispersing to the center of a spectral detector:
$n\lambda=4.76$ $\mathrm{nm}$ ($x=98.2$ mm of dispersion).

###  2.1 Optic Contributions

The development of the OGRE optic assembly is led by NASA GSFC [2]. Since this
component is manufactured externally, a detailed error analysis of this
component is beyond the scope of this error budget; however, similar error
budgets for mono-crystalline optic assemblies have been developed for other
X-ray missions such as Lynx [9]. The presented error budget only considers the
final performance of a 60$\mathrm{\SIUnitSymbolDegree}$ azimuthal section of
the OGRE optic assembly in the dispersion and cross-dispersion directions.

Table 1: Errors induced into the LSF of the OGRE spectrometer from a $60$$\mathrm{\SIUnitSymbolDegree}$ azimuthal section of the OGRE optic assembly. Since the optic will be treated as an assembled unit, the only errors to consider are its performance in the dispersion direction (measured as a full-width at half-maximum; FWHM) and the cross-dispersion direction (measured as a half-power diameter; HPD). Error |  DoF | Requirement ($3\sigma$) | LSF Impact
---|---|---|---
$\mathrm{\SIUnitSymbolMicro}$m ($\pm$) |  arcsec ($\pm$) |  Disp. [$\mathrm{\SIUnitSymbolMicro}$m] |  X-Disp. [$\mathrm{\SIUnitSymbolMicro}$m]
PSF |  Disp. Dir. |  – |  1.5 |  25.4 |  –
Cross-Disp. Dir. |  – |  5.0 |  – |  84.8
RSS Total |  25.4 |  84.8

The performance requirements of the OGRE optic assembly are derived from the
current performance of mono-crystalline optic segments. Single paraboloid-
hyperboloid mirror pairs routinely achieve a point spread function (PSF) with
half-power diameter (HPD) of $<2$ arcsec [2]. Conservative estimates suggest
that this performance will degrade slightly to $\sim 3-5$ arcsec HPD when all
288 individual segments are aligned together. In the dispersion direction, a
$60$$\mathrm{\SIUnitSymbolDegree}$ azimuthal section of the optic assembly
with this performance is expected to perform at $<1.5$ arcsec full-width at
half-maximum (FWHM). Thus, the OGRE optic performance requirements for a
60$\mathrm{\SIUnitSymbolDegree}$ azimuthal section of the optic assembly that
feeds a single grating module have been set at $<1.5$ arcsec FWHM in the
dispersion direction and $<5$ arcsec HPD in the cross-dispersion direction. At
a focal length of $Z_{0}=3500$ mm – the focal length of the OGRE optic
assembly – these dispersion and cross-dispersion requirements correspond to
$<25.4$ $\mathrm{\SIUnitSymbolMicro}$m FWHM and $<84.8$
$\mathrm{\SIUnitSymbolMicro}$m HPD, respectively. These errors are summarized
in Table 1.

The finite optic performance is thus the first contribution impacting the LSF
of the OGRE spectrometer. If the only contribution to the LSF is the PSF from
this optic, the spectrometer will exactly reproduce the PSF at dispersion.
With the dispersion limited to $x\sim 98.2$ mm due to the size of the payload,
this equates to a maximum achievable spectral resolution of $R(x/\Delta
x)\approx 3860$ for the system. While this performance is well beyond the OGRE
spectral resolution goal of $R>2000$, there are many additional contributions
in the system that also impact to the LSF of the OGRE spectrometer.

###  2.2 Grating Contributions

Beyond the finite performance of the optic, additional grating-related errors
are introduced into the observed LSF of grating spectrometers. In OGRE, these
grating-related errors arise from five main sources: aberrations induced by
the diffraction geometry, the finite performance of the individual gratings
within the grating module, and three misalignments during the construction of
the grating module and the alignment of this grating module to a
60$\mathrm{\SIUnitSymbolDegree}$ azimuthal section of the OGRE optic assembly.
Each of these five error sources will be discussed in the following
subsections.

####  2.2.1 Diffraction Geometry Aberration

The first contribution to the OGRE LSF from the gratings in the spectrometer
comes from the diffraction geometry itself. While the goal of a reflection
grating spectrometer such as OGRE is to exactly reproduce the optic PSF at
dispersion, this rarely happens in practice. Aberrations are induced into the
system which blur the diffracted-order LSF and cause it to diverge from the
optic PSF. These aberrations can arise from several sources, including
diffraction-induced astigmatism and the sampling of a curved focal plane with
a flat detector. For OGRE, the grating positions were numerically optimized
such that these aberrations were eliminated in the dispersion direction at Fe
XVII ($\lambda=15.01$ $\mathrm{\SIUnitSymbolAngstrom}$). However, while these
aberrations were eliminated in the dispersion direction, the extent of the LSF
in the cross-dispersion direction grew slightly durin this numerical
optimization exercise; the LSF error in the cross-dispersion direction was
60.5 $\mathrm{\SIUnitSymbolMicro}$m HPD. A summary of this induced LSF error
is listed in Table 2 as “Diff. Aberration”.

Table 2: Grating-induced errors to the observed LSF of the OGRE spectrometer, including aberrations induced by the diffraction geometry, the finite resolution limit of the individual gratings within a module, grating-level alignment, stack-level alignment, and module-to-optic alignment. Shown are the $3\sigma$ level (99.7%) requirements for each error in all six degrees of freedom (DoF; if applicable) and the impact of the error in both the dispersion direction (measured as a full-width at half maximum [FWHM]) and the cross-dispersion direction (measured as a half-power diameter [HPD]). Only the maximum LSF impact values are reported for each error in this table. Error |  DoF | Requirement ($3\sigma$) | LSF Impact
---|---|---|---
|  $\mathrm{\SIUnitSymbolMicro}$m ($\pm$) |  arcsec ($\pm$) |  Disp. [$\mathrm{\SIUnitSymbolMicro}$m] |  X-Disp. [$\mathrm{\SIUnitSymbolMicro}$m]
Diff. Aberration |  – |  – |  – |  – |  60.5
Grat. Res. Limit |  – | $\text{R}=\text{4500}$ |  21.8 |  –
Grating Alignment Within Stack |  X |  127 |  – |  0.4 |  102.6
Y |  127 |  – |  0.8 |  –
Z |  127 |  – |  8.5 |  –
Pitch (X) |  – |  30 |  – |  489.9
Yaw (Y) |  – |  30 |  0.6 |  282.2
Roll (Z) |  – |  15 |  10.8 |  5.7
Stack Alignment Within Module |  X |  127 |  – |  1.0 |  179.4
Y |  254 |  – |  – |  551.9
Z |  127 |  – |  6.7 |  21.4
Pitch (X) |  – |  60 |  0.3 |  1910.8
Yaw (Y) |  – |  60 |  3.6 |  1130.7
Roll (Z) |  – |  30 |  31.0 |  62.6
Module Alignment To Optic Section |  X |  500 |  – |  1.0 |  10.4
Y |  500 |  – |  2.0 |  10.9
Z |  1000 |  – |  1.7 |  39.9
Pitch (X) |  – |  60 |  1.0 |  9.2
Yaw (Y) |  – |  60 |  2.1 |  13.8
Roll (Z) |  – |  120 |  1.6 |  4.7
RSS Total |  41.2 |  2367.9

####  2.2.2 Grating Performance

The second contribution to the LSF from the gratings is the aberration due to
the finite resolution of the gratings that form an OGRE grating module. The
OGRE spectrometer is designed to utilize gratings with a radial groove profile
and a groove period of 160 nm at a distance of 3300 mm from the hub of the
converging grooves [10]. However, the processes used to manufacture these
gratings will introduce errors that cause the manufactured groove pattern to
deviate slightly from the idealized groove pattern. This deviation is expected
to be a random process; therefore, the groove period at any given location on
a grating will have a Gaussian distribution about its nominal value. From Eq.
2, this Gaussian groove period error will manifest on the focal plane as a
Gaussian error in the dispersion direction. These Gaussian period errors do
not impact the cross-dispersion extent.

An OGRE grating prototype was tested for spectral resolution at the Max Planck
Institute for Extraterrestrial Physics’ PANTER X-ray Test Facility in an
attempt to measure groove-induced aberrations. Results from this testing
indicate that the OGRE grating prototype achieved a groove-induced spectral
resolution of $R_{g}>4500$ at the $\sim 94$% confidence level [10]. However,
this achieved spectral resolution was found to be limited by the measurement
limit of the assembled spectrometer. Thus, the true grating-induced
aberrations of the OGRE grating prototype could not be measured in this test
and could have been higher than this limit. Similar X-ray gratings produced
via electron-beam lithography for synchrotron applications ($d=500$ nm) were
measured to have groove period errors of $<0.01$ nm [11, 12]. If similar
groove period errors can be achieved for OGRE gratings with a period of
$d=160$ nm, these gratings could achieve $R(d/\Delta d)\gtrsim 16,000$.
However, since this limit has not been measured explicitly for OGRE-like
periods, a conservative requirement of $R>4500$ ($3\sigma$) has been adopted
for the grating-induced resolution in the LSF error budget. If each grating
performs at this level, the impact on the observed LSF in the dispersion
direction is 21.8 $\mathrm{\SIUnitSymbolMicro}$m FWHM. This error is listed in
Table 2 as the “Grat. Res. Limit”.

Figure 4: Left \- A single reflection grating in the OGRE spectrometer [13].
Each grating has three 2 mm wide ribs manufactured into the back of the
substrate which allow light to pass through the grating stack while still
maintaining the precisely manufactured wedge profile. Overlayed on this figure
are the degrees of freedom (DoF) for each reflection grating. The grooves for
each grating converge in the $-$ẑ direction to the spectrometer focal plane.
Right \- An OGRE grating stack containing 30 individual reflection gratings
stacked on top of one another.

####  2.2.3 Grating Alignment

With the grating pattern manufactured, the construction of an OGRE grating
module can begin. Individual OGRE gratings start as wedged silicon substrates.
These substrates are manufactured such that their wedge angle replicates the
fan angle between neighboring gratings in each stack as required by the OGRE
optical design [1]. Each substrate will then be imprinted with the OGRE
grating pattern ($70$ mm x $70$ mm) via substrate conformal imprint
lithography [14]. The grating pattern will then be precisely diced from the
wedged substrate such that the sides of each grating are aligned relative to
the grating pattern itself. In addition to the dicing process, the majority of
the backside of each grating substrate will be removed leaving a face sheet
with a thickness of $\sim 0.3-0.5$ mm and three 2 mm wide x 70 mm long “ribs”
(depicted in Figure 4). These ribs allow light to pass through each OGRE
grating stack, but also maintain the precise wedge profile of each grating
within the stack. The backside of each grating substrate is then etched to
remove the stress introduced during this “ribbing” process. This completes the
manufacture of a grating substrate [1, 13].

Once individual grating substrates have been manufactured, they are then
stacked on top of each other to realize aligned grating stacks as depicted in
Figure 4. The wedged grating substrates themselves largely constrain the
grating-level pitch, roll, and ŷ alignment during this stacking process
(degrees of freedom shown in Figure 4). A precision robot will be used to
achieve alignment in the remaining three degrees of freedom (x̂, ẑ, and yaw)
by referencing the sides of each grating substrate and by maintaining a
precise global coordinate system during the stacking process. This alignment
methodology is similar to that utilized for the silicon pore optic (SPO)
technology developed by cosine Research B.V. [15] – a collaborator on these
alignment efforts.

The optical design of the spectrometer gives the desired placement of these
gratings within the two stacks. However, the manufacture of each grating and
the assembly of the 60 gratings into the two grating stacks will result in
placement errors of each grating relative to their designed placement. These
misalignments introduce aberrations into the observed LSF which will affect
the achievable performance of the OGRE spectrometer. The impact of a grating-
level misalignment in each degree of freedom is discussed below.

  * x̂: The grating pattern will be diced from the grating substrate to an accuracy of $<2$ $\mathrm{\SIUnitSymbolMicro}$m over the 70 mm length of the grating pattern. The stacking robot will then place this grating substrate into the stack. The precision of the stacking robot is $\lesssim 30$ $\mathrm{\SIUnitSymbolMicro}$m, but the interface between the robot and the grating substrate is currently unknown. In addition to the stacking robot itself, this interface is another source of error in the stacking process. Since this interface is currently unknown, the alignment tolerance in this degree of freedom has been set to $\pm 127$ $\mathrm{\SIUnitSymbolMicro}$m ($3\sigma$; standard machine tolerance). A misalignment at this level mainly impacts the cross-dispersion extent of the observed LSF with an contribution of $102.6$ $\mathrm{\SIUnitSymbolMicro}$m HPD, but has a small effect on the dispersion direction as well ($0.4$ $\mathrm{\SIUnitSymbolMicro}$m FWHM).

  * ŷ: Alignment in this degree of freedom is largely constrained by wedge manufacture. The manufacturer can easily meet standard machine tolerances in this degree of freedom, so this tolerance level has been adopted for this degree of freedom: $\pm 127$ $\mathrm{\SIUnitSymbolMicro}$m ($3\sigma$). A ŷ misalignment at this level only has a slight contribution to the LSF extent in the dispersion direction: $0.8$ $\mathrm{\SIUnitSymbolMicro}$m FWHM, but does not impact the LSF in the cross-dispersion direction.

  * ẑ: Just as with x̂ alignment, alignment in this degree of freedom is achieved largely by the stacking robot. Since the exact interface between the robot and the grating substrate is unknown, this tolerance has been set to $\pm 127$ $\mathrm{\SIUnitSymbolMicro}$m ($3\sigma$; standard machine tolerance). A ẑ misalignment at this level does not affect the observed LSF in the cross-dispersion direction, but has a significant impact on the dispersion direction: $8.5$ $\mathrm{\SIUnitSymbolMicro}$m FWHM.

  * Pitch (rotation about x̂): A pitch misalignment of an individual grating relative to its nominal orientation acts to move the diffraction arc in the cross-dispersion direction. Misalignments of all 60 gratings in pitch will then increase the extent of the combined LSF in the cross-dispersion dimension. The wedge manufacturer can achieve a tolerance on the wedge angle of $\pm 30$ arcsec ($3\sigma$), so this tolerance has been adopted as the grating-level pitch alignment tolerance. A pitch misalignment of each grating at this level increases the extent of the LSF in the cross-dispersion direction by $489.9$ $\mathrm{\SIUnitSymbolMicro}$m HPD.

  * Yaw (rotation about ŷ): A yaw misalignment of the gratings within the grating stacks will increase the extent of the LSF in the cross-dispersion direction. The dicing process has an accuracy of $<2$ $\mathrm{\SIUnitSymbolMicro}$m over the 70 mm length of the grating such that the grating edges will be aligned to $<6$ arcsec relative to the grating pattern itself. To limit the cross-dispersion impact on the LSF, the yaw alignment tolerance is $\pm 30$ arcsec ($3\sigma$). A yaw misalignment at this level will increase the extent of the LSF in the cross-dispersion direction by $282.2$ $\mathrm{\SIUnitSymbolMicro}$m HPD and will slightly increase the extent in the dispersion direction by 0.6 $\mathrm{\SIUnitSymbolMicro}$m FWHM.

  * Roll (rotation about ẑ): This degree of freedom is constrained both by wedge manufacture and the dicing process. The wedges can be manufactured such that the top and bottom surfaces are misaligned in roll by no more that $\pm 15$ arcsec. The grating pattern will then be aligned to the wedge direction to within $\pm 0.5$$\mathrm{\SIUnitSymbolDegree}$. Once aligned, the grating pattern is then diced to within $<2$ $\mathrm{\SIUnitSymbolMicro}$m over the 70 mm length of the grating. With the grating pattern aligned to within $\sim 0.5$$\mathrm{\SIUnitSymbolDegree}$ of the wedge direction, there is no additional roll induced by a misalignment of the wedged substrate and the grating pattern. Therefore, the roll alignment tolerance is set to that which is achievable during wedge manufacture: $\pm 15$ arcsec ($3\sigma$). A misalignment at this level increases the extent of the LSF in the dispersion direction by $10.8$ $\mathrm{\SIUnitSymbolMicro}$m FWHM and slightly increases the extent in the cross-dispersion direction by $5.7$ $\mathrm{\SIUnitSymbolMicro}$m HPD.

As mentioned previously, the interface between the stacking robot and the
grating substrates is currently unknown. Therefore, there are some
uncertainties in the achievable alignment of the gratings within stacks. While
x̂ and ẑ alignment tolerances were purposefully set to standard machine
tolerances to account for this unknown interface, the yaw tolerance is much
tighter than can be achieved by standard machine tolerances. Further
investigation will be needed to determine the interface between the grating
substrates and the stacking robot, and if an additional constraint mechanism
is needed. Precision pins are currently being investigated to serve as this
additional constraint. Since the edges of the gratings will be aligned to $<6$
arcsec relative to the grating pattern itself from the dicing process, each
grating side can reference two precision pins to constrain grating yaw within
a stack. Additionally, while the error budget baselines the wedged grating
alignment methodology presented here and in Donovan et al. (2019) [16], this
error budget could easily be adapted for other grating alignment methods if
needed.

####  2.2.4 Stack Alignment

Once the grating stacks have been constructed, they must be aligned into the
OGRE spectrometer. Rather than directly aligning the two stacks relative to
the optic, the stacks will first be integrated into a grating module. This
greatly eases spectrometer integration, but allows for additional
misalignments to be introduced into the spectrometer and therefore additional
aberrations to be introduced into the observed LSF.

Figure 5: The anticipated alignment methodology for the two grating stacks to
form a single grating module [13]. The two stacks would be placed on the
bottom surface, then abutted against Pins A & B and either Pin C or Pin D. The
precisely machined bottom plate constrains the ŷ, pitch, and roll of the
grating stacks, while pins constrain the remaining three degrees of freedom –
x̂, ẑ, and yaw. Here, Pins A & B constrain the x̂ and yaw alignment and Pins C
& D constrain the ẑ alignment between grating stacks.

Just as with grating-level alignment, the two stacks must be aligned to each
other in all six degrees of freedom. The currently envisioned stack-level
alignment methodology utilizes a precisely polished surface and additional
constraint pins to align the two stacks relative to one another [13]. This
system is depicted in Figure 5. The polished surface (polished to within $\pm
1$ $\mathrm{\SIUnitSymbolMicro}$m) orients the two stacks with respect to one
another in ŷ, pitch, and roll, while the pins constrain the stacks in the
three remaining degrees of freedom. The sides of grating stacks will be
abutted against Pins A & B to constrain the two stacks in x̂ and yaw, and the
back of each grating stack will be abutted against either Pin C or Pin D to
constrain the two stacks in the ẑ direction. A more complete discussion of
this alignment method can be found in O’Meara et al. (2019) [13]. Each degree
of freedom for stack-level alignment (as defined in Figure 5) and their impact
on the LSF is discussed below.

  * x̂: A misalignment in this degree of freedom acts to separate the LSFs formed by each grating stack in the cross-dispersion direction. Additionally, it has a small impact on the width of the LSF in the dispersion direction. Due to a weak dependence on both the dispersion and cross-dispersion extent, the x̂ alignment tolerance has been set to the standard machine tolerance: $\pm 127$ micron ($3\sigma$). This allows the two pins that constrain the two stacks in x̂ (Pin A & Pin B) to be placed using standard machining techniques. A misalignment of $\pm 127$ micron ($3\sigma$) contributes a dispersion extent of $1.0$ $\mathrm{\SIUnitSymbolMicro}$m FWHM and a cross-dispersion extent of $\sim 50-180$ $\mathrm{\SIUnitSymbolMicro}$m HPD with the exact impact depending on the specifics of the relative stack-to-stack misalignment.

  * ŷ: A misalignment in this degree of freedom will separate the LSFs from each stack in the cross-dispersion direction. Because the wedged substrates are manufactured with a ŷ tolerance of $\pm 127$ $\mathrm{\SIUnitSymbolMicro}$m ($3\sigma$), the maximum misalignment between the two stacks is $\pm 254$ $\mathrm{\SIUnitSymbolMicro}$m ($3\sigma$). A small error will also be introduced from the grating module base itself. However, this error ($\pm 1$ $\mathrm{\SIUnitSymbolMicro}$m) is negligible compared to the $\pm 254$ $\mathrm{\SIUnitSymbolMicro}$m ($3\sigma$) uncertainty from wedged substrate manufacture. Therefore, the alignment tolerance in this degree of freedom will be $\pm 254$ $\mathrm{\SIUnitSymbolMicro}$m ($3\sigma$). This ŷ misalignment will increase the total extent of the LSF in the cross-dispersion direction by $\sim 430-552$ $\mathrm{\SIUnitSymbolMicro}$m HPD.

  * ẑ: A misalignment in this degree of freedom will change the dispersion between the two stacks slightly such that the LSFs from each grating stack are dispersed to slightly different locations in the dispersion direction. This will increase the total extent of the combined LSF in the dispersion direction. This alignment will be constrained by pins (Pins C & D in Figure 5). Therefore, to ensure that this constraint can be placed using standard machining, a standard machine tolerance has been adopted in this degree of freedom: $\pm 127$ $\mathrm{\SIUnitSymbolMicro}$m ($3\sigma$). A misalignment at this level will increase the extent of the LSF in the dispersion direction by $\sim 3.6-6.7$ $\mathrm{\SIUnitSymbolMicro}$m FWHM, while only slightly impacting the extent in the cross-dispersion direction.

  * Pitch (rotation about x̂): A pitch of one grating stack (Grating Stack #1) relative to the other grating stack (Grating Stack #2) will cause the LSF formed by Grating Stack #1 to separate from the LSF formed by Grating Stack #2 in the cross-dispersion direction. This will increase the extent of the combined LSF formed by the two grating stacks in the cross-dispersion direction. The relative pitch between stacks is governed by both the bottom grating in each grating stack and the flatness of the polished base. With a pitch tolerance on each grating substrate of $\pm 30$ arcsec ($3\sigma$), the maximum misalignment between grating stacks induced by the stacks themselves is $\pm 60$ arcsec ($3\sigma$). O’Meara et al. (2019) show that the expected worst-case misalignment induced by the polished base is $\pm 12.7$ arcsec per grating stack. Thus, a total base-induced misalignment between the two grating stacks could be up to $25.4$ arcsec[13]. However, O’Meara et al. (2019) argue that the scenario assumed for this analysis is highly improbable and the likely base-induced misalignment is far below this value. Thus, it is assumed in this error budget that the grating-induced stack misalignment dominates the stack-level pitch misalignment, so an alignment tolerance of $\pm 60$ arcsec ($3\sigma$) has been adopted for this error. A pitch misalignment at this level will cause the combined LSF formed by the two grating stacks to increase in the cross-dispersion direction by $\sim 1800-1910$ $\mathrm{\SIUnitSymbolMicro}$m HPD.

  * Yaw (rotation about ŷ): Similar to a pitch misalignment between the two grating stacks, a yaw misalignment between the grating stacks will cause their individual LSFs to separate in the cross-dispersion direction. This separation will then cause the extent of the combined LSF in the cross-dispersion direction to grow. To limit the extent of the combined LSF in the cross-dispersion direction, this alignment tolerance has been set to $\pm 60$ arcsec ($3\sigma$). This leads to a $\sim 1020-1130$ $\mathrm{\SIUnitSymbolMicro}$m HPD growth in the cross-dispersion extent of the combined LSF. O’Meara et al. (2019) show that this alignment tolerance is achievable by abutting the two grating stacks against a set of precision pins (Pins A & B in Figure 5) [13].

  * Roll (rotation about ẑ): The misalignment of the two grating stacks in roll will cause the individual LSFs (which are much narrower in the dispersion direction when compared to the cross-dispersion direction) to tilt with respect to the mean dispersion direction of the two grating stacks. This will cause the combined LSF to have a greater extent in the dispersion direction. As with stack-level pitch, the stack-to-stack roll alignment is constrained by both the bottom grating in the grating stack and the polished base of the grating module. The grating substrates themselves have a roll requirement of $\pm 15$ arcsec ($3\sigma$), which results in a maximum stack misalignment of $\pm 30$ arcsec ($3\sigma$). Additionally, the base can contribute a misalignment of $12.6$ arcsec per stack for a total base-induced roll misalignment of $25.2$ arcsec as derived in O’Meara et al. (2019)[13]. However, just as with stack-level pitch, this base-induced misalignment is highly improbable and will likely be much lower than this value. Thus, a $\pm 30$ arcsec ($3\sigma$) has been adopted as the alignment tolerance in roll. This misalignment results in a LSF impact of $26.9-31.0$ $\mathrm{\SIUnitSymbolMicro}$m FWHM in the dispersion direction.

A summary of the derived stack-level tolerances can be seen in Table 2. For
stack-to-stack alignment, pitch and yaw misalignments have the largest impact
on the extent of the LSF in the cross-dispersion direction, while
misalignments in ẑ and roll have the largest impacts on the extent in the
dispersion direction.

####  2.2.5 Module Alignment

With a grating module fully assembled, it then must be aligned to its
designated 60 degree optic section. Compared to the errors from grating-level
and stack-level misalignments though, errors due to misalignments between the
grating module and the optic do not contribute significantly to the observed
LSF extent in the dispersion and cross-dispersion directions. Instead, these
errors shift the LSF centroid on the focal plane. As long as the important
spectral lines remain on the detector, this allows module-level alignment
tolerances to be slightly looser when compared to grating-level and stack-
level alignment tolerances as shown in Table 2.

Misalignments in x̂, ŷ, pitch, and yaw all shift the LSF centroid in the
cross-dispersion direction. A movement of $<\pm 1$ mm in the cross-dispersion
direction has been adopted as the maximum movement that the LSF centroid can
be shifted for each of these misalignments. The size of the detector is
$\sim\>$25 x 25 $\mathrm{m}\mathrm{m}^{2}$, so even if all four misalignments
contributed maximially, the movement of the LSF centroid would be limited to
$\sim 4$ mm. To limit the movement in this dimension to $\pm 1$ mm, x̂ and ŷ
misalignments have each been given a tolerance of $\pm 500$
$\mathrm{\SIUnitSymbolMicro}$m ($3\sigma$). They contribute insignificantly to
the extent of the LSF in the dispersion and cross-dispersion directions.
Similarly, pitch and yaw misalignments each have a tolerance of $\pm 60$
arcsec ($3\sigma$) to limit their cross-dispersion movements to $<\pm 1$ mm.
Pitch and yaw misalignments at this level also do not contribute significantly
to the LSF extent in either the dispersion or cross-dispersion directions.

A misalignment in roll contributes directly to a movement of the LSF centroid
in the dispersion direction, since the dispersion direction will change with
the roll of the grating module. The goal when assigning an alignment tolerance
to this degree of freedom is to keep important lines in the soft X-ray
spectrum of Capella on the detector. To satisfy this requirement, the roll
tolerance of the grating module relative to the optic has been set to $\pm
120$ arcsec ($3\sigma$).

The remaining misalignment to be considered is a ẑ misalignment. At the
millimeter level, this misalignment only changes the dispersion relation (Eq.
2) on the focal plane. A large change in dispersion on the focal plane will
move important lines off the focal plane. A $\sim 1$ mm movement of the
grating module relative to the optic, however, does not have any appreciable
change in the dispersion on the focal plane. Therefore, the alignment
tolerance for this degree of freedom is somewhat arbitrary. The ẑ alignment
tolerance has been set to $\pm 1$ mm ($3\sigma$) – a value which should be
achievable through standard machine tolerances (even with the stack-up of
several interfaces manufactured to standard machine tolerances). A summary of
the module-to-optic misalignments and their impact on the observed LSF extent
can be seen in Table 2.

###  2.3 Forward Assembly Contributions

Figure 6: CAD rendering of the OGRE payload showing the large separation (3.5 m) between the forward assembly (comprised of the optic and grating assemblies) and the detector assembly. Shown in the bottom-right corner is the coordinate system referenced in Section 2.3. Table 3: Errors induced by a misalignment of the forward assembly (aligned OGRE mirror module + OGRE grating module) into the observed LSF of the OGRE spectrometer. Shown are the $3\sigma$ level (99.7%) requirements for each error in all six degrees of freedom (DoF) and the impact of the error in both the dispersion direction (measured as a full-width at half maximum [FWHM]) and the cross-dispersion direction (measured as a half-power diameter [HPD]). Only the maximum LSF impact values are reported for each error in this table. Error |  DoF | Requirement ($3\sigma$) | LSF Impact
---|---|---|---
|  $\mathrm{\SIUnitSymbolMicro}$m ($\pm$) |  arcsec ($\pm$) |  Disp. [$\mathrm{\SIUnitSymbolMicro}$m] |  X-Disp. [$\mathrm{\SIUnitSymbolMicro}$m]
Forward Assembly To Nominal Focal Plane |  X |  600 |  – |  – |  –
Y |  1000 |  – |  – |  –
Z |  500 |  – |  14.8 |  –
Pitch (X) |  – |  60 |  – |  –
Yaw (Y) |  – |  90 |  7.9 |  –
Roll (Z) |  – |  1000 |  – |  –
RSS Total |  16.8 |  –

The aligned grating modules and optic assembly become the “forward assembly”.
This forward assembly must be aligned to the nominal focal plane. Whereas the
grating-level, stack-level, and module-to-optic alignment tolerances concern
components within close proximity to one another, the forward assembly is
$\sim 3.5$ m away from the nominal focal plane. This introduces new challenges
that must be considered when assigning alignment tolerances to this component.
Each degree of freedom (as depicted in Figure 6) for this alignment is
discussed below.

  * x̂: A misalignment in this degree of freedom moves the forward assembly in the dispersion direction relative to the nominal focal plane. This in turn moves the observed LSF on the spectral detectors in the dispersion direction. To keep important lines in the soft X-ray spectrum of Capella on the detector, the x̂ alignment tolerance has been set to $\pm 600$ $\mathrm{\SIUnitSymbolMicro}$m ($3\sigma$).

  * ŷ: A misalignment here causes the forward assembly to move in the cross-dispersion direction relative to the nominal focal plane, which also causes the observed LSF to move in the cross-dispersion direction on the spectral detectors. Similar to other LSF centroid translational errors, an alignment tolerance is chosen to limit the movement of the observed LSF on the focal plane to $<\pm 1$ mm. This equates to an alignment tolerance of $\pm 1000$ $\mathrm{\SIUnitSymbolMicro}$m ($3\sigma$).

  * ẑ: A shift in the forward assembly relative to the nominal focal plane in this dimension moves the observed LSF from its nominal focus position. This shift acts to defocus the spectrometer, broadening the LSF extent in the dispersion direction. To limit the impact on the extent of the LSF in the dispersion direction, while taking into consideration the practicality of aligning two components over a $\sim 3.5$ m distance, this alignment tolerance has been set to $\pm 500$ $\mathrm{\SIUnitSymbolMicro}$m ($3\sigma$). A misalignment at this level increases the LSF extent in the dispersion direction by $13.5-14.8$ $\mathrm{\SIUnitSymbolMicro}$m FWHM.

  * Pitch (rotation about x̂): A pitch of the forward assembly relative to the nominal focal plane acts to move the LSF in the cross-dispersion direction. To limit the translation in this direction to $<\pm 1$ mm, an alignment tolerance of $\pm 60$ arcsec ($3\sigma$) has been budgeted to this degree of freedom.

  * Yaw (rotation about ŷ): Similar to a yaw misalignment between the grating module and the optic, a yaw misalignment of the forward assembly relative to the nominal focal plane will move the centroid of the observed LSF in the cross-dispersion direction. This movement was limited to $<\pm 1$ mm ($3\sigma$), which corresponds to an alignment tolerance of $\pm 90$ arcsec. This has a slight impact on the extent of the LSF in the dispersion direction as well ($\sim 4.6-7.9$ $\mathrm{\SIUnitSymbolMicro}$m FWHM).

  * Roll (rotation about ẑ): A roll misalignment of the forward assembly relative to the nominal focal plane moves the observed LSF in the dispersion direction. To keep important spectral lines on the detector, this tolerance has been set to $\pm 0.28$$\mathrm{\SIUnitSymbolDegree}$ ($=\pm 1000$ arcsec).

The forward assembly will be mounted onto an optical bench cantilevered off of
a mounting point close to the focal plane of the spectrometer. To achieve the
derived alignment tolerances (as summarized in Table 3), an adjustable
kinematic mount will be designed as an interface between the forward assembly
and the optical bench. This mount would provide the forward assembly with
movement in ẑ, pitch, and yaw – the degrees of freedom with the tightest
alignment tolerances.

A concern with the alignment of the forward assembly is maintaining this
alignment during flight. Typically, this optical bench is made from several
cylindrical aluminum sections. However, it is anticipated that this aluminum
optical bench cannot maintain the derived alignment tolerances due to
potential thermal gradients over the length of the bench which would cause an
expansion/contraction of the optical bench. To better constrain the forward
assembly orientation relative to the nominal focal plane, a custom rigid
optical bench will be investigated.

###  2.4 Detector Contributions

The last misalignment in the OGRE spectrometer to consider is the alignment of
the spectral detectors relative to the nominal spectrometer focal plane. Just
as with all other potential misalignments, this component can be misaligned in
all six degrees of freedom. A discussion of each degree of freedom (as
depicted in Figure 7) and the impact of a misalignment in each of these
degrees of freedom is discussed below.

Figure 7: A CAD rendering of the OGRE focal plane and a single OGRE spectral detector. Left \- The OGRE focal plane with three spectral detectors and a central detector. The spectral detectors each sample diffraction arcs from two OGRE grating modules (geometry discussed in Figure 2), while the central detector samples the small fraction of light that does not interact with the six OGRE grating modules. Right \- A zoomed-in view of an OGRE spectral detector (outlined by the red box in the left-hand image). Shown in this image is the coordinate system for this particular spectral detector. Table 4: Errors introduced into the observed LSF of the OGRE spectrometer by a misalignment of a spectral detector relative to the nominal focal plane . Shown are the $3\sigma$ level (99.7%) requirements for each error in all six degrees of freedom (DoF) and the impact of the error in both the dispersion direction (measured as a full-width at half maximum [FWHM]) and the cross-dispersion direction (measured as a half-power diameter [HPD]). Only the maximum LSF impact values are reported for each error in this table. Error |  DoF | Requirement ($3\sigma$) | LSF Impact
---|---|---|---
|  $\mathrm{\SIUnitSymbolMicro}$m ($\pm$) |  arcsec ($\pm$) |  Disp. [$\mathrm{\SIUnitSymbolMicro}$m] |  X-Disp. [$\mathrm{\SIUnitSymbolMicro}$m]
Detector To Nominal Focal Plane |  X |  600 |  – |  – |  –
Y |  254 |  – |  – |  –
Z |  254 |  – |  8.0 |  –
Pitch (X) |  – |  3600 |  – |  –
Yaw (Y) |  – |  3600 |  – |  –
Roll (Z) |  – |  3600 |  – |  –
RSS Total |  8.0 |  –

  * x̂: A misalignment in this degree of freedom causes a shift of the detector in the dispersion direction relative to the nominal focal plane. If shifted too far, important spectral lines from Capella will begin to fall off of the detector. To keep important lines on the detector, an alignment tolerance of $\pm 600$ $\mathrm{\SIUnitSymbolMicro}$m has been assigned to this degree of freedom.

  * ŷ: A misalignment here causes the detector to move in the cross-dispersion direction relative to the nominal focal plane. As a result of this movement, the observed LSF will move on the detector. However, the observed LSF just needs to remain on the detector. An alignment tolerance of $\pm 254$ $\mathrm{\SIUnitSymbolMicro}$m has been assigned here, but realistically this could be loosened if necessary.

  * ẑ: A shift in ẑ of the detector relative to the nominal focal plane moves the observed LSF away from its nominal focus position. A misalignment here then acts to defocus the spectrometer which introduces aberrations into the LSF in the dispersion direction. To limit the impact on the extent of the observed LSF in the dispersion direction, this alignment tolerance has been set to $\pm 254$ $\mathrm{\SIUnitSymbolMicro}$m. A misalignment at this level increases the LSF extent in the dispersion direction by $7.2-8.0$ $\mathrm{\SIUnitSymbolMicro}$m FWHM.

  * Pitch (rotation about x̂): A pitch of the detector (about its center) relative to the nominal focal plane will act to defocus the LSF – one half of the LSF will be intrafocal and the other half will be extrafocal. However, only relatively large pitches ($>2$$\mathrm{\SIUnitSymbolDegree}$) will cause any appreciable impact on the extent of the observed LSF. Therefore, a relatively loose tolerance of $\pm 1$$\mathrm{\SIUnitSymbolDegree}$ has been adopted here. This tolerance can be loosened if necessary.

  * Yaw (rotation about ŷ): Similar to a misalignment of the detector in pitch, a yaw misalignment of the detector (about its center) relative to the nominal focal plane will move the portions of the spectrum on the detector out of its nominal focal position. Similar to pitch though, only large yaws ($>2$$\mathrm{\SIUnitSymbolDegree}$) will lead to any appreciable impact on the observed LSF. Therefore, the same $\pm 1$$\mathrm{\SIUnitSymbolDegree}$ alignment tolerance has been adopted here, but this could be loosened if necessary.

  * Roll (rotation about ẑ): A roll of the detector relative to the nominal focal plane will have no impact on the observed LSF. Its alignment tolerance has been set to $\pm 1$$\mathrm{\SIUnitSymbolDegree}$$=3600$ arcsec, but this tolerance is somewhat arbitrary.

As shown in Table 4, detector contributions only impact the extent of the
observed LSF at the $\sim 8$ $\mathrm{\SIUnitSymbolMicro}$m level. Compared to
contributions from the grating and the optic, misalignments of the detector do
not impact the LSF extent to any appreciable degree, only serving to move the
LSF centroid on the detector.

One complication in the alignment of the detectors is that their placement
within their packaging from the manufacturer (e2v) is not known to the level
of the derived alignment tolerances. Therefore, their placement must be
reconstructed once assembled and then adjusted to move them into their optimal
location within the alignment tolerances. Since the detectors cannot be
touched by traditional measuring devices such as a portable measuring arm or a
coordinate-measuring machine, a non-contact measuring method is required. The
baseline method for this non-contact measurement is a laser scanner attachment
to a portable measuring arm. With an accuracy of $\sim 40$
$\mathrm{\SIUnitSymbolMicro}$m, this laser scanner would be able to measure
each detector within its packaging to the required accuracy. With its position
and orientation known with respect to its packaging, each detector could then
be adjusted to place it in its proper location to within the required
alignment tolerances.

###  2.5 In-Flight Contributions: Jitter

The final contributor to the LSF of the OGRE spectrometer is the in-flight
contribution from jitter – the pointing stability of the payload during
flight. The frequency of this movement is expected to be higher than the
readout cadence of the detector, so the jitter of the payload about the source
will artificially increase the size of the source. This therefore impacts the
extent of the observed LSF in both the dispersion and cross-dispersion
directions.

The NASA Sounding Rockets User Handbook [17] states that the NSROC (NASA
Sounding Rocket Operations Contract) Celestial Attitude Control System can
achieve a jitter of $<1$ arcsec/sec FWHM in its linear thrust configuration
(the configuration baselined for the OGRE mission). This number has also been
confirmed from past sounding rocket flight data. Therefore, this number has
been adopted as the jitter requirement for the OGRE spectrometer. This
requirement corresponds to an induced aberration of 17
$\mathrm{\SIUnitSymbolMicro}$m FWHM in the dispersion direction and 14.4
$\mathrm{\SIUnitSymbolMicro}$m HPD in the cross-dispersion direction of the
LSF.

##  3 Combining Errors

With all performance requirements and alignment tolerances derived, they can
be combined to inform the achievable performance of the OGRE spectrometer.
Errors were first analyzed as if they were independent and all contributed at
their $3\sigma$ values, then raytrace simulations of the OGRE spectrometer
with these same errors and misalignments were performed to compare results.
Misalignments were then randomized within their $3\sigma$ limits and further
raytrace simulations were performed to understand spectrometer performance
when misalignments do not all contribute at their $3\sigma$ limits.

Table 5: All errors contributing to the observed LSF of the OGRE spectrometer, including errors from the optic assembly, grating module, forward assembly (aligned optic + grating module), detector, and in-flight contributions. Listed is each error and its contribution to the observed LSF in both the dispersion direction (measured as a full-width at half maximum [FWHM]) and the cross-dispersion direction (measured as a half-power diameter [HPD]).  Component | LSF Impact
---|---
Disp. [$\mathrm{\SIUnitSymbolMicro}$m] |  X-Disp. [$\mathrm{\SIUnitSymbolMicro}$m]
Optic |  25.4 |  84.8
Grating |  41.2 |  2367.9
Forward Assembly |  16.8 |  –
Detector |  8.0 |  –
Jitter |  17.0 |  14.4
RSS Total |  54.6 |  2369.5

###  3.1 The RSS Method: Independent Errors

If each induced error to the LSF is asssumed to be independent with respect to
the other induced errors, LSF errors add in quadrature (root sum of the
squares; RSS). In Tables 1-4, the individual errors from each component were
added in quadrature with their total LSF impact (“RSS Total”) in both the
dispersion and cross-dispersion directions displayed at the bottom of each
table. These totals were collected and are displayed together in Table 5. The
totals from each component were then added in quadrature to yield a total LSF
impact in both the dispersion and cross-dispersion directions. If all errors
contribute maximally, the resulting total impact on the observed LSF is 54.6
$\mathrm{\SIUnitSymbolMicro}$m FWHM in the dispersion direction and $\sim
2370$ $\mathrm{\SIUnitSymbolMicro}$m HPD in the cross-dispersion direction.
Converting the extent of the LSF in the dispersion direction ($\Delta x$) to
spectral resolution ($R=x/\Delta x$ with $x=98.2$ mm), the maximum achievable
spectral resolution if all LSF errors were at their $3\sigma$ limit and added
in quadrature is $R\approx 1800$. This result is below the spectral resolution
goal of $R>2000$, but it still comfortably meets the resolution requirement of
$R>1500$.

###  3.2 Raytrace Simulation Results

To test whether the assumption that all errors are independent and add in
quadrature is a fair assumption, raytrace simulations of the OGRE spectrometer
were performed with all misalignments and errors at their $3\sigma$ limits
(the $3\sigma$ requirements presented in Tables 1-4). A total of
$N_{sim}=1000$ simulations of the OGRE spectrometer were performed with these
misalignments and errors at the blaze wavelength of the system:
$n\lambda=4.76$ nm. For each simulation, the spectral resolution at the blaze
wavelength was calculated. The distribution of the resulting resolution values
attained from simulations is presented in Figure 8a. Further, the result of an
individual simulation is shown in Figure 8b. The results from this raytrace
simulation are consistent with the results obtained when assuming all errors
are independent and add in quadrature – $R\approx 1880$ (median value). So
therefore, if all errors contributed at their $3\sigma$ error limits, the OGRE
spectrometer would not achieve its spectral resolution goal of $R>2000$, but
would still comfortably achieve its spectral resolution requirement of
$R>1500$ at this wavelength.

---
(a) (b)
Figure 8:  Results from 1,000 raytrace simulations of the OGRE spectrometer
with misalignments, performance errors, and jitter at their $3\sigma$ limits
(presented in Tables 1-4). (a): Distribution of the achieved spectral
resolutions from the 1,000 raytrace simulations. These values range from
$R\sim 1790$ ($2\sigma$ low) to $R\sim 1980$ ($2\sigma$ high), with a median
spectral resolution of $R=1880$. (b): The diffracted-order LSF from a single
raytrace simulation in (a). Structure within this LSF comes from the two
misaligned grating stacks (two groups centered at $x\sim-0.02$ and $x\sim
0.02$ in the dispersion direction) and also from the individual gratings
within each stack (quasi-horizontal lines within the two grating stack
groups). This diffracted-order LSF achieves a spectral resolution of $R\sim
1880$.

It should be noted that diffracted-order LSF resulting from a single
simulation of the OGRE spectrometer in Figure 8b demonstrates structure; it is
apparent that two separate LSFs make up this combined LSF. This structure
suggest that the LSFs formed by the individual grating stacks could
potentially be used to provide a higher resolution spectra, since the
individual LSFs are much narrower than the combined LSF. Identification and
subsequent analysis of these individual LSFs that make up the combined LSF
would require thorough calibration of the instrument prior to launch to ensure
all four LSFs on a single spectral detector – two from this grating module and
another two from a second grating module clocked 60 degrees relative to the
first (geometry shown in Figure 2) – can be individiually identified. The
potential for this analysis will be investigated in the future.

The previous simulations assumed all misalignments and performance errors
contributed at their $3\sigma$ error limit. In reality though, some
misalignments will be well within their $3\sigma$ limit, while others might be
closer to the periphery of their limits. To understand how this situation
might improve the achievable resolution of the OGRE spectrometer, further
raytrace simulations were performed. However, instead of assuming that all
contributions to the LSF were at their $3\sigma$ limits, all misalignments
were given random values within their $3\sigma$ parameter space. Performance
contributions (optic performance, grating resolution limit, and jitter) were
kept at their $3\sigma$ limits, since values for these contributions are
expected to be closer to their $3\sigma$ limits. Similar to the previous
simulations, this raytrace simulation was performed $N_{sim}=1000$ times with
each simulation having unique misalignment values. The results of these
simulations are presented in Figure 9. These simulations show that while there
are still some incarnations of the spectrometer that do not achieve the
$R>2000$ performance goal ($<2.5$% of simulations), the vast majority of
spectrometers modelled achieved a spectral resolution beyond this goal
($>97.5$%).

Figure 9: Results from 1,000 raytrace simulations of the OGRE spectrometer
with misalignment values chosen randomly within their $3\sigma$ requirements,
but with performance contributions (optic performance, grating period errors,
and jitter) still contributing at their $3\sigma$ error limits. The achieved
spectral resolutions range from $R\sim 2000$ ($2\sigma$ low) to $R\sim 2990$
($2\sigma$ high), with a median spectral resolution of $R=2700$.

##  4 SUMMARY

In this manuscript, a comprehensive LSF error budget for the soft X-ray
grating spectrometer on the Off-plane Grating Rocket Experiment (OGRE) was
described. This error budget described potential impacts to the LSF observed
by the OGRE spectrometer including component misalignments and performance
errors, derived realistic alignment tolerances for component misalignments,
and elucidated how each misalignment and performance error impacted the
observed LSF. The impacts from component misalignments and performance errors
were combined first by assuming independent errors such that the errors added
in quadrature. It was found that if all errors contribute at their $3\sigma$
error limits, the spectrometer would achieve a spectral resolution of $\sim
1800-1900$. This performance meets the OGRE spectral resolution requirement of
$R>1500$ comfortably, but does not meet the spectral resolution goal of
$R>2000$. However, if not all misalignments contributed at their $3\sigma$
error limits, but instead were randomly distributed within their $\pm 3\sigma$
error limits, the spectrometer meets the spectral resolution goal of $R>2000$
for $>97.5$% of the simulations. These results suggest that the OGRE
spectrometer should be able to comfortably meet its spectral resolution
requirement of $R>1500$, and, depending on the exact values of misalignments
during assembly, even meet its spectral resolution goal of $R>2000$.

### Disclosures

The authors have no relevant financial interests in the manuscript and no
other potential conflicts of interest to disclose.

###### Acknowledgements.

The work presented in this manuscript is supported by NASA grant NNX17AD19G, a
NASA Space Technology Research Fellowship, and internal funding from The
Pennsylvania State University. The authors would like to thank current and
past members of the McEntaffer research group for their support of this
project. This work makes use of PyXFocus, an open-source Python-based raytrace
package.

## References

  * [1] B. D. Donovan, R. L. McEntaffer, J. H. Tutt, et al., “An updated optical design of the off-plane grating rocket experiment,” Proc. SPIE 11119, 1111911 (2019).
  * [2] W. W. Zhang, K. D. Allgood, M. P. Biskach, et al., “Next generation x-ray optics for astronomy: high resolution, lightweight, and low cost,” Proc. SPIE 11119, 1111907 (2019).
  * [3] R. McEntaffer, C. DeRoo, T. Schultz, et al., “First results from a next-generation off-plane X-ray diffraction grating,” Experimental Astronomy 36(1), 389–405 (2013).
  * [4] M. R. F. Lewis, M. R. Soman, A. D. Holland, et al., “Development of the x-ray camera for the OGRE sub-orbital rocket,” Proc. SPIE 9915, 991506 (2016).
  * [5] H. Wolter, “Spiegelsysteme streifenden einfalls als abbildende optiken für röntgenstrahlen,” Ann. Phys. 445(1-2), 94–114 (1952).
  * [6] D. M. Miles, J. A. McCoy, R. L. McEntaffer, et al., “Fabrication and Diffraction Efficiency of a Large-format, Replicated X-Ray Reflection Grating,” ApJ 869(2), 95 (2018).
  * [7] W. C. Cash, “X-ray optics. 2: A technique for high resolution spectroscopy,” Appl. Opt. 30(13), 1749–1759 (1991).
  * [8] W. C. Cash, “X-ray spectrographs using radial groove gratings,” Appl. Opt. 22(24), 3971–3976 (1983).
  * [9] W. W. Zhang, K. D. Allgood, M. P. Biskach, et al., “High-resolution, lightweight, and low-cost x-ray optics for the Lynx observatory,” Journal of Astronomical Telescopes, Instruments, and Systems 5, 021012 (2019).
  * [10] B. D. Donovan, R. L. McEntaffer, C. T. DeRoo, et al., “Performance Testing of a Large-Format X-ray Reflection Grating Prototype for a Suborbital Rocket Payload,” Journal of Astronomical Instrumentation 9(4), 2050017 (2020).
  * [11] D. L. Voronov, E. M. Gullikson, and H. A. Padmore, “Large area nanoimprint enables ultra-precise x-ray diffraction gratings,” Opt. Express 25(19), 23334–23342 (2017).
  * [12] C. T. DeRoo, R. L. McEntaffer, B. D. Donovan, et al., “Large-format X-Ray Reflection Grating Operated in an Echelle-like Mounting,” ApJ 897(1), 92 (2020).
  * [13] B. C. O’Meara, B. D. Donovan, J. H. Tutt, et al., “The opto-mechanical design for the Off-plane Grating Rocket Experiment,” Proc. SPIE 11100, 1110006 (2019).
  * [14] M. A. Verschuuren, M. Megens, Y. Ni, et al., “Large area nanoimprint by substrate conformal imprint lithography (SCIL),” Advanced Optical Technologies 6, 243–264 (2017).
  * [15] M. J. Collon, G. Vacanti, N. Barriere, et al., “Silicon pore optics mirror module production and testing,” Proc. SPIE 10699, 106990Y (2018).
  * [16] B. D. Donovan, R. L. McEntaffer, J. H. Tutt, et al., “A comprehensive line spread function error budget for the Off-plane Grating Rocket Experiment,” Proc. SPIE 11119, 1111912 (2019).
  * [17] NASA Goddard Space Flight Center - Sounding Rockets Program Office, “NASA Sounding Rockets User Handbook.”

Benjamin D. Donovan is a PhD candidate at The Pennsylvania State University in
the Department of Astronomy & Astrophysics. He received his B.S. in Physics &
Astronomy from The University of Iowa in 2016, and his M.S. in Astronomy &
Astrophysics from The Pennsylvania State University in 2019. He is the
recipient of a NASA Space Technology Research Fellowship. His current research
interests include the design and development of space-based astronomical
instrumentation. He is a member of SPIE.

Biographies and photographs of the other authors are not available.

###### List of Figures

  1. 1 The diffraction geometry of a reflection grating in the extreme off-plane mount [6, 7]. Light is incident from point A onto the grating surface at a graze angle $\eta$ and relative to the groove direction by an angle $\Psi$. Equivalently, this incidence geometry can be described in the spherical coordinate system by azimuth angle $\alpha$ and polar angle $\gamma$. Diffraction follows the generalized grating equation (Eq. 1) and light is diffracted a distance $L$ to azimuth angle $\beta$ on the focal plane. The dispersion distance between the $n=0$ reflection and $n$-th diffraction order on this focal plane is given by $x=n\lambda L/d$.
  2. 2 Diffraction geometry for neighboring grating modules in the OGRE soft X-ray spectrometer [1]. The optic assembly is divided into six 60$\mathrm{\SIUnitSymbolDegree}$ azimuthal sections. Behind each of these sections is a grating module containing 60 gratings. To maximize diffraction efficiency, each grating module operates in the Littrow mount which requires $\alpha=\beta=\delta$. For the OGRE spectrometer, $\alpha=\beta=\delta=30$$\mathrm{\SIUnitSymbolDegree}$ such that the neighboring grating modules diffract to the same location on the focal plane. This geometry is repeated two times to populate the entire 360$\mathrm{\SIUnitSymbolDegree}$ azimuthal span of optic with grating modules. Depicted in the bottom-right is the coordinate system for this schematic view with +ẑ pointing into the page.
  3. 3 Results from a raytrace simulation of the OGRE spectrometer showing the effect of a yaw misalignment of individual gratings in the grating module on the extent of the OGRE LSF in the dispersion (blue) and cross-dispersion (orange) directions. The contribution in the dispersion direction is measured as a full-width at half-maximum (FWHM), while the contribution in the cross-dispersion direction is measured as a half-power diameter (HPD). Here, it can been seen that a grating-level yaw misalignment at these simulated misalignment values does not contribute to the LSF extent in the dispersion direction, but it has a noticeable impact on the LSF extent in the cross-dispersion direction.
  4. 4 Left \- A single reflection grating in the OGRE spectrometer [13]. Each grating has three 2 mm wide ribs manufactured into the back of the substrate which allow light to pass through the grating stack while still maintaining the precisely manufactured wedge profile. Overlayed on this figure are the degrees of freedom (DoF) for each reflection grating. The grooves for each grating converge in the $-$ẑ direction to the spectrometer focal plane. Right \- An OGRE grating stack containing 30 individual reflection gratings stacked on top of one another.
  5. 5 The anticipated alignment methodology for the two grating stacks to form a single grating module [13]. The two stacks would be placed on the bottom surface, then abutted against Pins A & B and either Pin C or Pin D. The precisely machined bottom plate constrains the ŷ, pitch, and roll of the grating stacks, while pins constrain the remaining three degrees of freedom – x̂, ẑ, and yaw. Here, Pins A & B constrain the x̂ and yaw alignment and Pins C & D constrain the ẑ alignment between grating stacks.
  6. 6 CAD rendering of the OGRE payload showing the large separation (3.5 m) between the forward assembly (comprised of the optic and grating assemblies) and the detector assembly. Shown in the bottom-right corner is the coordinate system referenced in Section 2.3.
  7. 7 A CAD rendering of the OGRE focal plane and a single OGRE spectral detector. Left \- The OGRE focal plane with three spectral detectors and a central detector. The spectral detectors each sample diffraction arcs from two OGRE grating modules (geometry discussed in Figure 2), while the central detector samples the small fraction of light that does not interact with the six OGRE grating modules. Right \- A zoomed-in view of an OGRE spectral detector (outlined by the red box in the left-hand image). Shown in this image is the coordinate system for this particular spectral detector.
  8. 8  Results from 1,000 raytrace simulations of the OGRE spectrometer with misalignments, performance errors, and jitter at their $3\sigma$ limits (presented in Tables 1-4). (a): Distribution of the achieved spectral resolutions from the 1,000 raytrace simulations. These values range from $R\sim 1790$ ($2\sigma$ low) to $R\sim 1980$ ($2\sigma$ high), with a median spectral resolution of $R=1880$. (b): The diffracted-order LSF from a single raytrace simulation in (a). Structure within this LSF comes from the two misaligned grating stacks (two groups centered at $x\sim-0.02$ and $x\sim 0.02$ in the dispersion direction) and also from the individual gratings within each stack (quasi-horizontal lines within the two grating stack groups). This diffracted-order LSF achieves a spectral resolution of $R\sim 1880$.
  9. 9 Results from 1,000 raytrace simulations of the OGRE spectrometer with misalignment values chosen randomly within their $3\sigma$ requirements, but with performance contributions (optic performance, grating period errors, and jitter) still contributing at their $3\sigma$ error limits. The achieved spectral resolutions range from $R\sim 2000$ ($2\sigma$ low) to $R\sim 2990$ ($2\sigma$ high), with a median spectral resolution of $R=2700$.

###### List of Tables

  1. 1 Errors induced into the LSF of the OGRE spectrometer from a $60$$\mathrm{\SIUnitSymbolDegree}$ azimuthal section of the OGRE optic assembly. Since the optic will be treated as an assembled unit, the only errors to consider are its performance in the dispersion direction (measured as a full-width at half-maximum; FWHM) and the cross-dispersion direction (measured as a half-power diameter; HPD).
  2. 2 Grating-induced errors to the observed LSF of the OGRE spectrometer, including aberrations induced by the diffraction geometry, the finite resolution limit of the individual gratings within a module, grating-level alignment, stack-level alignment, and module-to-optic alignment. Shown are the $3\sigma$ level (99.7%) requirements for each error in all six degrees of freedom (DoF; if applicable) and the impact of the error in both the dispersion direction (measured as a full-width at half maximum [FWHM]) and the cross-dispersion direction (measured as a half-power diameter [HPD]). Only the maximum LSF impact values are reported for each error in this table.
  3. 3 Errors induced by a misalignment of the forward assembly (aligned OGRE mirror module + OGRE grating module) into the observed LSF of the OGRE spectrometer. Shown are the $3\sigma$ level (99.7%) requirements for each error in all six degrees of freedom (DoF) and the impact of the error in both the dispersion direction (measured as a full-width at half maximum [FWHM]) and the cross-dispersion direction (measured as a half-power diameter [HPD]). Only the maximum LSF impact values are reported for each error in this table.
  4. 4 Errors introduced into the observed LSF of the OGRE spectrometer by a misalignment of a spectral detector relative to the nominal focal plane . Shown are the $3\sigma$ level (99.7%) requirements for each error in all six degrees of freedom (DoF) and the impact of the error in both the dispersion direction (measured as a full-width at half maximum [FWHM]) and the cross-dispersion direction (measured as a half-power diameter [HPD]). Only the maximum LSF impact values are reported for each error in this table.
  5. 5 All errors contributing to the observed LSF of the OGRE spectrometer, including errors from the optic assembly, grating module, forward assembly (aligned optic + grating module), detector, and in-flight contributions. Listed is each error and its contribution to the observed LSF in both the dispersion direction (measured as a full-width at half maximum [FWHM]) and the cross-dispersion direction (measured as a half-power diameter [HPD]).