A Comprehensive Line-Spread Function Error Budget for the Off-Plane Grating Rocket Experiment
Benjamin D. Donovan, Randall L. McEntaffer, James H. Tutt, Bridget C. O'Meara, Fabien Grisé, William W. Zhang, Michael P. Biskach, Timo T. Saha, Andrew D. Holland, Daniel Evan, Matthew R. Lewis, Matthew R. Soman, Karen Holland, David Colebrook, Fraser Cooper, David Farn
AA Comprehensive Line-Spread Function Error Budget for theOff-Plane Grating Rocket Experiment
Benjamin D. Donovan a,* , Randall L. McEntaffer a , James H. Tutt a , Bridget C. O’Meara a ,Fabien Gris´e a , William W. Zhang b , Michael P. Biskach b , Timo T. Saha b , Andrew D.Holland c , Daniel Evan c , Matthew R. Lewis c , Matthew R. Soman c , Karen Holland d , DavidColebrook d , Fraser Cooper d , David Farn d a The Pennsylvania State University, University Park, PA 16802 b NASA Goddard Space Flight Center, Greenbelt, MD 20771 c The Open University, Walton Hall, Milton Keynes, UK d XCAM Ltd., Northampton, UK
Abstract.
The Off-plane Grating Rocket Experiment (OGRE) is a soft X-ray grating spectrometer to be flown ona suborbital rocket. The payload is designed to obtain the highest-resolution soft X-ray spectrum of Capella to datewith a resolution goal of R( λ/ ∆ λ ) > at select wavelengths in its 10–55 ˚A bandpass of interest. The opticaldesign of the spectrometer realizes a theoretical maximum resolution of R ≈ , but this performance does notconsider the finite performance of the individual spectrometer components, misalignments between components, andin-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 identifycontributions to the LSF, to determine how each of these affects the LSF, and to inform performance requirementsand alignment tolerances for the spectrometer. In this document, the comprehensive LSF error budget for the OGREspectrometer is presented, the resulting errors are validated via raytrace simulations, and the implications of theseresults are discussed. Keywords: error budget, X-ray spectroscopy, suborbital rocket, reflection gratings, mono-crystalline silicon X-rayoptics, electron-multiplying CCDs. * Benjamin D. Donovan, [email protected]
The Off-plane Grating Rocket Experiment (OGRE) is a soft X-ray grating spectrometer that willbe flown on a suborbital rocket. With a spectral resolution requirement of R ( λ/ ∆ λ ) > across its 10 – 55 ˚A bandpass of interest and a goal of R > at select wavelengths in thissame 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 ( α Auriga), inunprecedented detail. This detailed observation will permit existing line blends in the soft X-rayspectrum of Capella to be resolved, new and updated emission lines to be integrated into plasmaspectral models, and more accurate plasma characteristics to be determined for this source. To achieve its performance goal of
R > , OGRE will utilize three cutting-edge technolo-gies: a mono-crystalline silicon X-ray optic assembly manufactured by NASA Goddard SpaceFlight Center (GSFC), six reflection grating modules developed by The Pennsylvania State Uni-versity operated in the extreme off-plane mount and integrated into a grating assembly, and anarray of four electron-multiplying CCDs (EM-CCDs) manufactured by e2v and integrated into adetector assembly by XCAM Ltd. and The Open University. a r X i v : . [ a s t r o - ph . I M ] J a n igure 1 The diffraction geometry of a reflection grating in the extreme off-plane mount.
Light is incident frompoint A onto the grating surface at a graze angle η and relative to the groove direction by an angle Ψ . Equivalently,this incidence geometry can be described in the spherical coordinate system by azimuth angle α and polar angle γ .Diffraction follows the generalized grating equation (Eq. 1) and light is diffracted a distance L to azimuth angle β onthe focal plane. The dispersion distance between the n = 0 reflection and n -th diffraction order on this focal plane isgiven by x = nλL/d . The OGRE spectrometer employs the traditional three-component X-ray grating spectrometer de-sign consisting of an X-ray optic that focuses the incident light from Capella, a reflection gratingarray that diffracts this light into its component spectrum, and an array of detectors at the focalplane to sample the spectrum.The X-ray optic on board the OGRE spectrometer will be a mono-crystalline silicon X-ray opticassembly developed by NASA GSFC. The optic assembly will consist of twelve Wolter I-type (paraboloid + hyperboloid) mirror shells with radii from r = 162 . − . mm and a commonfocal length of Z = 3500 mm. Each of these shells will be constructed from individual siliconmirror segments that span 30° in azimuth. A more detailed description of this optic assembly canbe found in Donovan et al. (2019). The OGRE spectrometer will utilize reflection gratings operated in the extreme off-plane mountto 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 atgrazing incidence relative to the incident X-ray photons as depicted in Figure 1. Diffraction thenfollows the generalized grating equation: sin α + sin β = nλd sin γ , (1)where α is the incident azimuthal angle, β is the diffracted azimuthal angle, γ is the polar angle2etween the incident light and the groove direction, d is the groove period, n is the diffractionorder, and λ is the wavelength of the light. By differentiating Eq. 1 with respect to λ , it can beshown that: dλdx = 10 nLD ˚A mm , (2)where L is the distance a photon on the grating travels to the spectrometer focal plane, D is thegroove density ( ≡ /d ), and x is the distance a photon is diffracted from the n = 0 reflection( = L sin γ (sin β + sin α ) ). This equation shows that while the extreme off-plane mount diffractsits incident light conically, the spectral information is contained only in one dimension – the dis-persion direction ( x in Figure 1).In the OGRE spectrometer, the full 360° azimuthal span of the optic assembly is divided into60° azimuthal sections. Behind each of these azimuthal sections is a OGRE grating module con-taining 60 individual reflection gratings organized into two side-by-side grating stacks. The gratingpositions in each module are numerically optimized to realize maximum spectral resolution at theFe XVII emission line ( λ = 15 . ˚A) – the brightest line expected to be observed from Capella.Light from two neighboring grating modules is diffracted to the same location on the focal planewhere the spectra are read out by a single detector. A schematic of this diffraction geometry can beseen in Figure 2. Spectral isolation on this detector will be attained through a combination of thedetector’s energy resolution and a slight offset ( ∼ − mm) of the diffraction arcs relative to oneanother. This geometry is repeated two additional times around the optic assembly for a total of sixgrating modules diffracting to three separate spectral detectors (also shown in Figure 2). Furtherdiscussion of this geometry can be found in Donovan et al. (2019). The optical design of the OGRE spectrometer realizes a maximum spectral resolution of R ≈ at the Fe XVII emission line. This resolution is only attained though if all spectrometercomponents 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 measuredperformance of the gratings in the OGRE grating modules and the mirror segments in the OGREoptic assembly will deviate from their idealized performance. These performance errors will beginto degrade the diffracted line-spread function (LSF) of the spectrometer as observed on the focalplane. Furthermore, the spectrometer components will not be aligned perfectly to one other. Thesemisalignments will further decrease the achievable performance of the spectrometer. Finally, thespectrometer will not remain perfectly pointed at Capella during its observation, but will insteaddither about its ideal pointing. This in-flight jitter will further degrade the observed LSF. Thesethree sources of error will all conspire to cause the spectrometer to not achieve its theoreticalspectral resolution of R ≈ , but to achieve a performance below this level.To understand how each misalignment and error contributes to the performance of the OGREspectrometer, a comprehensive LSF error budget is required. This error budget identifies eachcontribution 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 OGREspectrometer so that the spectrometer can meet its overall performance requirement of R > .In this manuscript, the comprehensive LSF error budget for OGRE is described and implicationsresulting from this error budget are discussed. 3 ptic Focus Shared Diffraction ArcZero Order (Module 𝛼𝛼 𝛽𝛽 𝛼𝛼 𝛽𝛽 Shared DetectorDispersion Direction (Module
Figure 2
Diffraction geometry for neighboring grating modules in the OGRE soft X-ray spectrometer. The opticassembly is divided into six 60° azimuthal sections. Behind each of these sections is a grating module containing60 gratings. To maximize diffraction efficiency, each grating module operates in the Littrow mount which requires α = β = δ . For the OGRE spectrometer, α = β = δ = 30 ° such that the neighboring grating modules diffract tothe same location on the focal plane. This geometry is repeated two times to populate the entire 360° azimuthal spanof optic with grating modules. Depicted in the bottom-right is the coordinate system for this schematic view with +ˆzpointing into the page. The comprehensive LSF error budget considers potential misalignments and performance errorsin the spectrometer. These misalignments and performance errors come from each of the threemain 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 ofthe payload. Each of these misalignments and performance errors modifies the observed LSFfrom 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 thespectrometer. Additionally, misalignments can shift the centroid of the LSF on the detector. Theseshifts can move important spectral lines of Capella off of the detector, impacting the science returnof the spectrometer. Thus, each potential performance error and misalignment must be analyzedto ensure the OGRE performance requirements are met.The construction of the LSF error budget begins with the consideration of a single contribution.For example, consider the yaw misalignment (rotation about ˆy; degrees of freedom defined inFigure 2) of the 60 gratings within the two grating stacks that form Grating Module igure 3
Results from a raytrace simulation of the OGRE spectrometer showing the effect of a yaw misalignmentof 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 doesnot contribute to the LSF extent in the dispersion direction, but it has a noticeable impact on the LSF extent in thecross-dispersion direction. final LSF of the spectrometer, a custom raytrace model a of the OGRE spectrometer is utilized.A range of potential yaw misalignment values are simulated in this spectrometer model and theresulting LSF is measured for each. The results of these simulations are shown in Figure 3. In thisfigure, it can be seen that a yaw misalignment of the gratings within the two grating stacks affectsthe extent of the OGRE LSF in the cross-dispersion direction, but has no impact on the extentof the LSF in the dispersion direction. This agrees with what is expected from theory. Based onthese results, a yaw alignment tolerance for the gratings within a stack would then be set (to bediscussed in Section 2.2.3). Similar simulations would then be run for the remaining five degreesof freedom ( ˆx, ˆy, ˆz, pitch, and roll) of the gratings within their stacks, and requirements for thesemisalignments would be derived as well. Further simulations would then be performed for allremaining misalignments and performance errors within the spectrometer.Throughout the construction of the error budget, the performance requirements and goals arecontinually referenced to ensure that the spectrometer will meet these requirements and goals. ForOGRE, the spectral resolution goal is R > (resolution requirement:
R > ) which limitsthe extent of the LSF in the dispersion direction. In the cross-dispersion direction, the entire LSFmust remain within the planned window size of the spectral detector ( ∼ mm). While the LSFmust remain within this window, movement of the LSF centroid in the cross-dispersion directioncan be larger than this since the window can be moved on the detector. Total movement is limited to ∼ ± mm since the LSF must still remain on the detector ( ∼ mm x mm; e2v CCD207-40 ).In the following subsections, individual contributions to the OGRE LSF error budget will bediscussed. The discussion here is limited to a single 60° azimuthal section of the OGRE optic a Based upon PyXFocus: https://github.com/rallured/PyXFocus able 1 Errors induced into the LSF of the OGRE spectrometer from a ° azimuthal section of the OGRE opticassembly. Since the optic will be treated as an assembled unit, the only errors to consider are its performance in thedispersion direction (measured as a full-width at half-maximum; FWHM) and the cross-dispersion direction (measuredas a half-power diameter; HPD). Error DoF Requirement ( σ ) LSF Impactµm ( ± ) arcsec ( ± ) Disp. [µm] X-Disp. [µm]PSF Disp. Dir. – 1.5 25.4 –Cross-Disp. Dir. – 5.0 – 84.8 RSS Total nλ = 4 .
76 nm ( x = 98 . mm ofdispersion). The development of the OGRE optic assembly is led by NASA GSFC. Since this componentis manufactured externally, a detailed error analysis of this component is beyond the scope ofthis error budget; however, similar error budgets for mono-crystalline optic assemblies have beendeveloped for other X-ray missions such as
Lynx . The presented error budget only considers thefinal performance of a 60° azimuthal section of the OGRE optic assembly in the dispersion andcross-dispersion directions.The performance requirements of the OGRE optic assembly are derived from the current per-formance of mono-crystalline optic segments. Single paraboloid-hyperboloid mirror pairs rou-tinely achieve a point spread function (PSF) with half-power diameter (HPD) of < arcsec. Conservative estimates suggest that this performance will degrade slightly to ∼ − arcsec HPDwhen all 288 individual segments are aligned together. In the dispersion direction, a ° azimuthalsection of the optic assembly with this performance is expected to perform at < . arcsec full-width at half-maximum (FWHM). Thus, the OGRE optic performance requirements for a 60°azimuthal section of the optic assembly that feeds a single grating module have been set at < . arcsec FWHM in the dispersion direction and < arcsec HPD in the cross-dispersion direction.At a focal length of Z = 3500 mm – the focal length of the OGRE optic assembly – these dis-persion and cross-dispersion requirements correspond to < . µm FWHM and < . µm HPD,respectively. These errors are summarized in Table 1.The finite optic performance is thus the first contribution impacting the LSF of the OGREspectrometer. If the only contribution to the LSF is the PSF from this optic, the spectrometer willexactly reproduce the PSF at dispersion. With the dispersion limited to x ∼ . mm due to thesize of the payload, this equates to a maximum achievable spectral resolution of R ( x/ ∆ x ) ≈ for the system. While this performance is well beyond the OGRE spectral resolution goal of6 > , there are many additional contributions in the system that also impact to the LSF of theOGRE spectrometer. Beyond the finite performance of the optic, additional grating-related errors are introduced into theobserved LSF of grating spectrometers. In OGRE, these grating-related errors arise from five mainsources: aberrations induced by the diffraction geometry, the finite performance of the individualgratings within the grating module, and three misalignments during the construction of the gratingmodule and the alignment of this grating module to a 60° azimuthal section of the OGRE opticassembly. Each of these five error sources will be discussed in the following subsections.
The first contribution to the OGRE LSF from the gratings in the spectrometer comes from thediffraction geometry itself. While the goal of a reflection grating spectrometer such as OGRE isto exactly reproduce the optic PSF at dispersion, this rarely happens in practice. Aberrations areinduced into the system which blur the diffracted-order LSF and cause it to diverge from the opticPSF. These aberrations can arise from several sources, including diffraction-induced astigmatismand the sampling of a curved focal plane with a flat detector. For OGRE, the grating positions werenumerically optimized such that these aberrations were eliminated in the dispersion direction at FeXVII ( λ = 15 . ˚A). 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 optimiza-tion exercise; the LSF error in the cross-dispersion direction was 60.5 µm HPD. A summary of thisinduced LSF error is listed in Table 2 as “Diff. Aberration”. The second contribution to the LSF from the gratings is the aberration due to the finite resolutionof the gratings that form an OGRE grating module. The OGRE spectrometer is designed to utilizegratings with a radial groove profile and a groove period of 160 nm at a distance of 3300 mmfrom the hub of the converging grooves. However, the processes used to manufacture thesegratings will introduce errors that cause the manufactured groove pattern to deviate slightly fromthe idealized groove pattern. This deviation is expected to be a random process; therefore, thegroove period at any given location on a grating will have a Gaussian distribution about its nominalvalue. From Eq. 2, this Gaussian groove period error will manifest on the focal plane as a Gaussianerror in the dispersion direction. These Gaussian period errors do not impact the cross-dispersionextent.An OGRE grating prototype was tested for spectral resolution at the Max Planck Institute forExtraterrestrial Physics’ PANTER X-ray Test Facility in an attempt to measure groove-inducedaberrations. Results from this testing indicate that the OGRE grating prototype achieved a groove-induced spectral resolution of R g > at the ∼ % confidence level. However, this achievedspectral resolution was found to be limited by the measurement limit of the assembled spectrom-eter. Thus, the true grating-induced aberrations of the OGRE grating prototype could not be mea-sured in this test and could have been higher than this limit. Similar X-ray gratings producedvia electron-beam lithography for synchrotron applications ( d = 500 nm) were measured to havegroove period errors of < . nm.
11, 12
If similar groove period errors can be achieved for OGRE7 able 2
Grating-induced errors to the observed LSF of the OGRE spectrometer, including aberrations induced by thediffraction 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 σ level (99.7%) requirements for each error inall six degrees of freedom (DoF; if applicable) and the impact of the error in both the dispersion direction (measuredas 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 ( σ ) LSF Impactµm ( ± ) arcsec ( ± ) Disp. [µm] X-Disp. [µm]Diff. Aberration – – – – 60.5Grat. Res. Limit – R = RSS Total igure 4
Left - A single reflection grating in the OGRE spectrometer. Each grating has three 2 mm wide ribsmanufactured into the back of the substrate which allow light to pass through the grating stack while still maintainingthe precisely manufactured wedge profile. Overlayed on this figure are the degrees of freedom (DoF) for each reflectiongrating. The grooves for each grating converge in the − ˆz direction to the spectrometer focal plane. Right - An OGREgrating stack containing 30 individual reflection gratings stacked on top of one another. gratings with a period of d = 160 nm, these gratings could achieve R ( d/ ∆ d ) (cid:38) , . However,since this limit has not been measured explicitly for OGRE-like periods, a conservative require-ment of R > ( σ ) has been adopted for the grating-induced resolution in the LSF errorbudget. If each grating performs at this level, the impact on the observed LSF in the dispersiondirection is 21.8 µm FWHM. This error is listed in Table 2 as the “Grat. Res. Limit”. 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 manufacturedsuch that their wedge angle replicates the fan angle between neighboring gratings in each stackas required by the OGRE optical design. Each substrate will then be imprinted with the OGREgrating pattern ( mm x mm) via substrate conformal imprint lithography. The gratingpattern will then be precisely diced from the wedged substrate such that the sides of each gratingare aligned relative to the grating pattern itself. In addition to the dicing process, the majority ofthe backside of each grating substrate will be removed leaving a face sheet with a thickness of ∼ . − . mm and three 2 mm wide x 70 mm long “ribs” (depicted in Figure 4). These ribsallow light to pass through each OGRE grating stack, but also maintain the precise wedge profileof each grating within the stack. The backside of each grating substrate is then etched to removethe stress introduced during this “ribbing” process. This completes the manufacture of a gratingsubstrate.
1, 13
Once individual grating substrates have been manufactured, they are then stacked on top ofeach other to realize aligned grating stacks as depicted in Figure 4. The wedged grating substrates9hemselves largely constrain the grating-level pitch, roll, and ˆy alignment during this stacking pro-cess (degrees of freedom shown in Figure 4). A precision robot will be used to achieve alignmentin the remaining three degrees of freedom (ˆx, ˆz, and yaw) by referencing the sides of each grat-ing 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) technologydeveloped by cosine Research B.V. – a collaborator on these alignment efforts.The optical design of the spectrometer gives the desired placement of these gratings withinthe two stacks. However, the manufacture of each grating and the assembly of the 60 gratingsinto the two grating stacks will result in placement errors of each grating relative to their designedplacement. These misalignments introduce aberrations into the observed LSF which will affect theachievable performance of the OGRE spectrometer. The impact of a grating-level misalignment ineach degree of freedom is discussed below. ˆx: The grating pattern will be diced from the grating substrate to an accuracy of < µmover the 70 mm length of the grating pattern. The stacking robot will then place this gratingsubstrate into the stack. The precision of the stacking robot is (cid:46) µm, but the interfacebetween the robot and the grating substrate is currently unknown. In addition to the stackingrobot itself, this interface is another source of error in the stacking process. Since this inter-face is currently unknown, the alignment tolerance in this degree of freedom has been set to ± µm ( σ ; standard machine tolerance). A misalignment at this level mainly impacts thecross-dispersion extent of the observed LSF with an contribution of . µm HPD, but hasa small effect on the dispersion direction as well ( . µm FWHM). ˆy: Alignment in this degree of freedom is largely constrained by wedge manufacture. Themanufacturer can easily meet standard machine tolerances in this degree of freedom, so thistolerance level has been adopted for this degree of freedom: ± µm ( σ ). A ˆy misalign-ment at this level only has a slight contribution to the LSF extent in the dispersion direction: . µm FWHM, but does not impact the LSF in the cross-dispersion direction. ˆz: Just as with ˆx alignment, alignment in this degree of freedom is achieved largely bythe stacking robot. Since the exact interface between the robot and the grating substrate isunknown, this tolerance has been set to ± µm ( σ ; standard machine tolerance). A ˆzmisalignment at this level does not affect the observed LSF in the cross-dispersion direction,but has a significant impact on the dispersion direction: . µm FWHM. Pitch (rotation about ˆx):
A pitch misalignment of an individual grating relative to its nom-inal orientation acts to move the diffraction arc in the cross-dispersion direction. Misalign-ments of all 60 gratings in pitch will then increase the extent of the combined LSF in thecross-dispersion dimension. The wedge manufacturer can achieve a tolerance on the wedgeangle of ± arcsec ( σ ), so this tolerance has been adopted as the grating-level pitch align-ment tolerance. A pitch misalignment of each grating at this level increases the extent of theLSF in the cross-dispersion direction by . µm HPD. Yaw (rotation about ˆy):
A yaw misalignment of the gratings within the grating stacks willincrease the extent of the LSF in the cross-dispersion direction. The dicing process has anaccuracy of < µm over the 70 mm length of the grating such that the grating edges will10e aligned to < arcsec relative to the grating pattern itself. To limit the cross-dispersionimpact on the LSF, the yaw alignment tolerance is ± arcsec ( σ ). A yaw misalignment atthis level will increase the extent of the LSF in the cross-dispersion direction by . µmHPD and will slightly increase the extent in the dispersion direction by 0.6 µm FWHM. Roll (rotation about ˆz):
This degree of freedom is constrained both by wedge manufactureand the dicing process. The wedges can be manufactured such that the top and bottomsurfaces are misaligned in roll by no more that ± arcsec. The grating pattern will thenbe aligned to the wedge direction to within ± . °. Once aligned, the grating pattern is thendiced to within < µm over the 70 mm length of the grating. With the grating pattern alignedto within ∼ . ° of the wedge direction, there is no additional roll induced by a misalignmentof the wedged substrate and the grating pattern. Therefore, the roll alignment tolerance is setto that which is achievable during wedge manufacture: ± arcsec ( σ ). A misalignment atthis level increases the extent of the LSF in the dispersion direction by . µm FWHM andslightly increases the extent in the cross-dispersion direction by . µm HPD.As mentioned previously, the interface between the stacking robot and the grating substrates is cur-rently unknown. Therefore, there are some uncertainties in the achievable alignment of the gratingswithin stacks. While ˆx and ˆz alignment tolerances were purposefully set to standard machine toler-ances to account for this unknown interface, the yaw tolerance is much tighter than can be achievedby standard machine tolerances. Further investigation will be needed to determine the interface be-tween the grating substrates and the stacking robot, and if an additional constraint mechanism isneeded. Precision pins are currently being investigated to serve as this additional constraint. Sincethe edges of the gratings will be aligned to < arcsec relative to the grating pattern itself from thedicing process, each grating side can reference two precision pins to constrain grating yaw withina stack. Additionally, while the error budget baselines the wedged grating alignment methodologypresented here and in Donovan et al. (2019), this error budget could easily be adapted for othergrating alignment methods if needed. Once the grating stacks have been constructed, they must be aligned into the OGRE spectrom-eter. Rather than directly aligning the two stacks relative to the optic, the stacks will first beintegrated into a grating module. This greatly eases spectrometer integration, but allows for addi-tional misalignments to be introduced into the spectrometer and therefore additional aberrations tobe introduced into the observed LSF.Just as with grating-level alignment, the two stacks must be aligned to each other in all six de-grees of freedom. The currently envisioned stack-level alignment methodology utilizes a preciselypolished surface and additional constraint pins to align the two stacks relative to one another. This system is depicted in Figure 5. The polished surface (polished to within ± µm) orients thetwo stacks with respect to one another in ˆy, pitch, and roll, while the pins constrain the stacks in thethree remaining degrees of freedom. The sides of grating stacks will be abutted against Pins A & Bto constrain the two stacks in ˆx and yaw, and the back of each grating stack will be abutted againsteither Pin C or Pin D to constrain the two stacks in the ˆz direction. A more complete discussionof this alignment method can be found in O’Meara et al. (2019). Each degree of freedom forstack-level alignment (as defined in Figure 5) and their impact on the LSF is discussed below.11 z Pin A Pin BPin DPin C y Figure 5
The anticipated alignment methodology for the two grating stacks to form a single grating module. The twostacks would be placed on the bottom surface, then abutted against Pins A & B and either Pin C or Pin D. The preciselymachined bottom plate constrains the ˆy, pitch, and roll of the grating stacks, while pins constrain the remaining threedegrees of freedom – ˆx, ˆz, and yaw. Here, Pins A & B constrain the ˆx and yaw alignment and Pins C & D constrainthe ˆz alignment between grating stacks. ˆx:
A misalignment in this degree of freedom acts to separate the LSFs formed by each gratingstack in the cross-dispersion direction. Additionally, it has a small impact on the widthof the LSF in the dispersion direction. Due to a weak dependence on both the dispersionand cross-dispersion extent, the ˆx alignment tolerance has been set to the standard machinetolerance: ± micron ( σ ). This allows the two pins that constrain the two stacks in ˆx(Pin A & Pin B) to be placed using standard machining techniques. A misalignment of ± micron ( σ ) contributes a dispersion extent of . µm FWHM and a cross-dispersionextent of ∼ − µm HPD with the exact impact depending on the specifics of the relativestack-to-stack misalignment. ˆy: A misalignment in this degree of freedom will separate the LSFs from each stack inthe cross-dispersion direction. Because the wedged substrates are manufactured with a ˆytolerance of ± µm ( σ ), the maximum misalignment between the two stacks is ± µm( σ ). A small error will also be introduced from the grating module base itself. However,this error ( ± µm) is negligible compared to the ± µm ( σ ) uncertainty from wedgedsubstrate manufacture. Therefore, the alignment tolerance in this degree of freedom will be ± µm ( σ ). This ˆy misalignment will increase the total extent of the LSF in the cross-dispersion direction by ∼ − µm HPD. ˆz: A misalignment in this degree of freedom will change the dispersion between the twostacks slightly such that the LSFs from each grating stack are dispersed to slightly differentlocations in the dispersion direction. This will increase the total extent of the combined12SF in the dispersion direction. This alignment will be constrained by pins (Pins C & D inFigure 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: ± µm ( σ ).A misalignment at this level will increase the extent of the LSF in the dispersion directionby ∼ . − . µm FWHM, while only slightly impacting the extent in the cross-dispersiondirection. Pitch (rotation about ˆx):
A pitch of one grating stack (Grating Stack ± arcsec ( σ ), the maximum misalignment between grating stacksinduced by the stacks themselves is ± arcsec ( σ ). O’Meara et al. (2019) show that theexpected worst-case misalignment induced by the polished base is ± . arcsec per gratingstack. Thus, a total base-induced misalignment between the two grating stacks could be upto . arcsec. However, O’Meara et al. (2019) argue that the scenario assumed for thisanalysis is highly improbable and the likely base-induced misalignment is far below thisvalue. Thus, it is assumed in this error budget that the grating-induced stack misalignmentdominates the stack-level pitch misalignment, so an alignment tolerance of ± arcsec ( σ )has been adopted for this error. A pitch misalignment at this level will cause the com-bined LSF formed by the two grating stacks to increase in the cross-dispersion direction by ∼ − µm HPD. Yaw (rotation about ˆy):
Similar to a pitch misalignment between the two grating stacks,a yaw misalignment between the grating stacks will cause their individual LSFs to separatein the cross-dispersion direction. This separation will then cause the extent of the combinedLSF in the cross-dispersion direction to grow. To limit the extent of the combined LSF inthe cross-dispersion direction, this alignment tolerance has been set to ± arcsec ( σ ). Thisleads to a ∼ − µm HPD growth in the cross-dispersion extent of the combinedLSF. O’Meara et al. (2019) show that this alignment tolerance is achievable by abutting thetwo grating stacks against a set of precision pins (Pins A & B in Figure 5). Roll (rotation about ˆz):
The misalignment of the two grating stacks in roll will cause theindividual LSFs (which are much narrower in the dispersion direction when compared tothe cross-dispersion direction) to tilt with respect to the mean dispersion direction of thetwo grating stacks. This will cause the combined LSF to have a greater extent in the dis-persion direction. As with stack-level pitch, the stack-to-stack roll alignment is constrainedby both the bottom grating in the grating stack and the polished base of the grating mod-ule. The grating substrates themselves have a roll requirement of ± arcsec ( σ ), whichresults in a maximum stack misalignment of ± arcsec ( σ ). Additionally, the base cancontribute a misalignment of . arcsec per stack for a total base-induced roll misalignmentof . arcsec as derived in O’Meara et al. (2019). However, just as with stack-level pitch,this base-induced misalignment is highly improbable and will likely be much lower than this13alue. Thus, a ± arcsec ( σ ) has been adopted as the alignment tolerance in roll. Thismisalignment results in a LSF impact of . − . µm FWHM in the dispersion direction.A summary of the derived stack-level tolerances can be seen in Table 2. For stack-to-stack align-ment, pitch and yaw misalignments have the largest impact on the extent of the LSF in the cross-dispersion direction, while misalignments in ˆz and roll have the largest impacts on the extent in thedispersion direction. With a grating module fully assembled, it then must be aligned to its designated 60 degree opticsection. Compared to the errors from grating-level and stack-level misalignments though, errorsdue to misalignments between the grating module and the optic do not contribute significantly tothe observed LSF extent in the dispersion and cross-dispersion directions. Instead, these errorsshift the LSF centroid on the focal plane. As long as the important spectral lines remain on thedetector, this allows module-level alignment tolerances to be slightly looser when compared tograting-level and stack-level alignment tolerances as shown in Table 2.Misalignments in ˆx, ˆy, pitch, and yaw all shift the LSF centroid in the cross-dispersion direc-tion. A movement of < ± mm in the cross-dispersion direction has been adopted as the maximummovement that the LSF centroid can be shifted for each of these misalignments. The size of thedetector is ∼
25 x 25 mm , so even if all four misalignments contributed maximially, the move-ment of the LSF centroid would be limited to ∼ mm. To limit the movement in this dimensionto ± mm, ˆx and ˆy misalignments have each been given a tolerance of ± µm ( σ ). They con-tribute insignificantly to the extent of the LSF in the dispersion and cross-dispersion directions.Similarly, pitch and yaw misalignments each have a tolerance of ± arcsec ( σ ) to limit theircross-dispersion movements to < ± mm. Pitch and yaw misalignments at this level also do notcontribute 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 dispersiondirection, since the dispersion direction will change with the roll of the grating module. The goalwhen assigning an alignment tolerance to this degree of freedom is to keep important lines in thesoft X-ray spectrum of Capella on the detector. To satisfy this requirement, the roll tolerance ofthe grating module relative to the optic has been set to ± arcsec ( σ ).The remaining misalignment to be considered is a ˆz misalignment. At the millimeter level, thismisalignment only changes the dispersion relation (Eq. 2) on the focal plane. A large change indispersion on the focal plane will move important lines off the focal plane. A ∼ mm movementof the grating module relative to the optic, however, does not have any appreciable change in thedispersion on the focal plane. Therefore, the alignment tolerance for this degree of freedom issomewhat arbitrary. The ˆz alignment tolerance has been set to ± mm ( σ ) – a value which shouldbe achievable through standard machine tolerances (even with the stack-up of several interfacesmanufactured to standard machine tolerances). A summary of the module-to-optic misalignmentsand their impact on the observed LSF extent can be seen in Table 2. The aligned grating modules and optic assembly become the “forward assembly”. This forwardassembly must be aligned to the nominal focal plane. Whereas the grating-level, stack-level, and14 .5 m Detector AssemblyForward Assembly Grating AssemblyOptic Assembly yz x 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 thecoordinate system referenced in Section 2.3.
Table 3
Errors induced by a misalignment of the forward assembly (aligned OGRE mirror module + OGRE gratingmodule) into the observed LSF of the OGRE spectrometer. Shown are the σ level (99.7%) requirements for eacherror in all six degrees of freedom (DoF) and the impact of the error in both the dispersion direction (measured as afull-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 ( σ ) LSF Impactµm ( ± ) arcsec ( ± ) Disp. [µm] X-Disp. [µm]ForwardAssembly ToNominalFocal Plane X 600 – – –Y 1000 – – –Z 500 – 14.8 –Pitch (X) – 60 – –Yaw (Y) – 90 7.9 –Roll (Z) – 1000 – – RSS Total ∼ . m away from the nominal focal plane. This introduces new chal-lenges that must be considered when assigning alignment tolerances to this component. Eachdegree of freedom (as depicted in Figure 6) for this alignment is discussed below. ˆx: A misalignment in this degree of freedom moves the forward assembly in the dispersiondirection relative to the nominal focal plane. This in turn moves the observed LSF on thespectral detectors in the dispersion direction. To keep important lines in the soft X-rayspectrum of Capella on the detector, the ˆx alignment tolerance has been set to ± µm( σ ). ˆy: A misalignment here causes the forward assembly to move in the cross-dispersion direc-tion relative to the nominal focal plane, which also causes the observed LSF to move in thecross-dispersion direction on the spectral detectors. Similar to other LSF centroid transla-tional errors, an alignment tolerance is chosen to limit the movement of the observed LSFon the focal plane to < ± mm. This equates to an alignment tolerance of ± µm ( σ ). ˆz: A shift in the forward assembly relative to the nominal focal plane in this dimensionmoves the observed LSF from its nominal focus position. This shift acts to defocus thespectrometer, broadening the LSF extent in the dispersion direction. To limit the impacton the extent of the LSF in the dispersion direction, while taking into consideration thepracticality of aligning two components over a ∼ . m distance, this alignment tolerancehas been set to ± µm ( σ ). A misalignment at this level increases the LSF extent in thedispersion direction by . − . µm FWHM. Pitch (rotation about ˆx):
A pitch of the forward assembly relative to the nominal focalplane acts to move the LSF in the cross-dispersion direction. To limit the translation in thisdirection to < ± mm, an alignment tolerance of ± arcsec ( σ ) has been budgeted to thisdegree of freedom. Yaw (rotation about ˆy):
Similar to a yaw misalignment between the grating module and theoptic, a yaw misalignment of the forward assembly relative to the nominal focal plane willmove the centroid of the observed LSF in the cross-dispersion direction. This movement waslimited to < ± mm ( σ ), which corresponds to an alignment tolerance of ± arcsec. Thishas a slight impact on the extent of the LSF in the dispersion direction as well ( ∼ . − . µmFWHM). Roll (rotation about ˆz):
A roll misalignment of the forward assembly relative to the nominalfocal plane moves the observed LSF in the dispersion direction. To keep important spectrallines on the detector, this tolerance has been set to ± . ° ( = ± arcsec).The forward assembly will be mounted onto an optical bench cantilevered off of a mounting pointclose to the focal plane of the spectrometer. To achieve the derived alignment tolerances (as sum-marized in Table 3), an adjustable kinematic mount will be designed as an interface between theforward assembly and the optical bench. This mount would provide the forward assembly withmovement in ˆz, pitch, and yaw – the degrees of freedom with the tightest alignment tolerances.16 x z Figure 7
A CAD rendering of the OGRE focal plane and a single OGRE spectral detector.
Left - The OGRE focalplane with three spectral detectors and a central detector. The spectral detectors each sample diffraction arcs from twoOGRE grating modules (geometry discussed in Figure 2), while the central detector samples the small fraction of lightthat 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 particularspectral detector.
A concern with the alignment of the forward assembly is maintaining this alignment duringflight. 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 tolerancesdue to potential thermal gradients over the length of the bench which would cause an expan-sion/contraction of the optical bench. To better constrain the forward assembly orientation relativeto the nominal focal plane, a custom rigid optical bench will be investigated.
The last misalignment in the OGRE spectrometer to consider is the alignment of the spectral detec-tors 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 offreedom (as depicted in Figure 7) and the impact of a misalignment in each of these degrees offreedom is discussed below. ˆx:
A misalignment in this degree of freedom causes a shift of the detector in the dispersiondirection relative to the nominal focal plane. If shifted too far, important spectral lines fromCapella will begin to fall off of the detector. To keep important lines on the detector, analignment tolerance of ± µm has been assigned to this degree of freedom. ˆy: A misalignment here causes the detector to move in the cross-dispersion direction relativeto the nominal focal plane. As a result of this movement, the observed LSF will move onthe detector. However, the observed LSF just needs to remain on the detector. An alignment17 able 4
Errors introduced into the observed LSF of the OGRE spectrometer by a misalignment of a spectral detectorrelative to the nominal focal plane . Shown are the σ level (99.7%) requirements for each error in all six degrees offreedom (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 LSFimpact values are reported for each error in this table. Error DoF Requirement ( σ ) LSF Impactµm ( ± ) arcsec ( ± ) Disp. [µm] X-Disp. [µm]Detector ToNominalFocal Plane X 600 – – –Y 254 – – –Z 254 – 8.0 –Pitch (X) – 3600 – –Yaw (Y) – 3600 – –Roll (Z) – 3600 – – RSS Total ± µm has been assigned here, but realistically this could be loosened ifnecessary. ˆz: A shift in ˆz of the detector relative to the nominal focal plane moves the observed LSFaway from its nominal focus position. A misalignment here then acts to defocus the spec-trometer which introduces aberrations into the LSF in the dispersion direction. To limit theimpact on the extent of the observed LSF in the dispersion direction, this alignment toler-ance has been set to ± µm. A misalignment at this level increases the LSF extent in thedispersion direction by . − . µm FWHM. Pitch (rotation about ˆx):
A pitch of the detector (about its center) relative to the nominalfocal plane will act to defocus the LSF – one half of the LSF will be intrafocal and theother half will be extrafocal. However, only relatively large pitches ( > °) will cause anyappreciable impact on the extent of the observed LSF. Therefore, a relatively loose toleranceof ± ° has been adopted here. This tolerance can be loosened if necessary. Yaw (rotation about ˆy):
Similar to a misalignment of the detector in pitch, a yaw mis-alignment of the detector (about its center) relative to the nominal focal plane will movethe portions of the spectrum on the detector out of its nominal focal position. Similar topitch though, only large yaws ( > °) will lead to any appreciable impact on the observedLSF. Therefore, the same ± ° alignment tolerance has been adopted here, but this could beloosened if necessary. Roll (rotation about ˆz):
A roll of the detector relative to the nominal focal plane will haveno impact on the observed LSF. Its alignment tolerance has been set to ± ° = 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 ∼ µm level. Compared to contributions from the grating and the optic, misalignments of the18etector do not impact the LSF extent to any appreciable degree, only serving to move the LSFcentroid on the detector.One complication in the alignment of the detectors is that their placement within their pack-aging 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 theminto their optimal location within the alignment tolerances. Since the detectors cannot be touchedby traditional measuring devices such as a portable measuring arm or a coordinate-measuring ma-chine, a non-contact measuring method is required. The baseline method for this non-contactmeasurement is a laser scanner attachment to a portable measuring arm. With an accuracy of ∼ µm, this laser scanner would be able to measure each detector within its packaging to the requiredaccuracy. With its position and orientation known with respect to its packaging, each detectorcould then be adjusted to place it in its proper location to within the required alignment tolerances. 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 expectedto be higher than the readout cadence of the detector, so the jitter of the payload about the sourcewill artificially increase the size of the source. This therefore impacts the extent of the observedLSF in both the dispersion and cross-dispersion directions.The NASA Sounding Rockets User Handbook states that the NSROC (NASA SoundingRocket Operations Contract) Celestial Attitude Control System can achieve a jitter of < arc-sec/sec FWHM in its linear thrust configuration (the configuration baselined for the OGRE mis-sion). This number has also been confirmed from past sounding rocket flight data. Therefore, thisnumber has been adopted as the jitter requirement for the OGRE spectrometer. This requirementcorresponds to an induced aberration of 17 µm FWHM in the dispersion direction and 14.4 µmHPD in the cross-dispersion direction of the LSF. With all performance requirements and alignment tolerances derived, they can be combined toinform the achievable performance of the OGRE spectrometer. Errors were first analyzed as ifthey were independent and all contributed at their σ values, then raytrace simulations of theOGRE spectrometer with these same errors and misalignments were performed to compare results.Misalignments were then randomized within their σ limits and further raytrace simulations wereperformed to understand spectrometer performance when misalignments do not all contribute attheir σ limits. If each induced error to the LSF is asssumed to be independent with respect to the other inducederrors, LSF errors add in quadrature (root sum of the squares; RSS). In Tables 1-4, the individualerrors 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. Thesetotals were collected and are displayed together in Table 5. The totals from each component werethen added in quadrature to yield a total LSF impact in both the dispersion and cross-dispersiondirections. If all errors contribute maximally, the resulting total impact on the observed LSF is19 able 5
All errors contributing to the observed LSF of the OGRE spectrometer, including errors from the opticassembly, 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-widthat half maximum [FWHM]) and the cross-dispersion direction (measured as a half-power diameter [HPD]).
Component LSF ImpactDisp. [µm] X-Disp. [µm]Optic 25.4 84.8Grating 41.2 2367.9Forward Assembly 16.8 –Detector 8.0 –Jitter 17.0 14.4
RSS Total ∼ µm HPD in the cross-dispersion direction.Converting the extent of the LSF in the dispersion direction ( ∆ x ) to spectral resolution ( R = x/ ∆ x with x = 98 . mm), the maximum achievable spectral resolution if all LSF errors were at their σ limit and added in quadrature is R ≈ . This result is below the spectral resolution goal of R > , but it still comfortably meets the resolution requirement of
R > . To test whether the assumption that all errors are independent and add in quadrature is a fair as-sumption, raytrace simulations of the OGRE spectrometer were performed with all misalignmentsand errors at their σ limits (the σ 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 theblaze wavelength of the system: nλ = 4 . nm. For each simulation, the spectral resolution at theblaze wavelength was calculated. The distribution of the resulting resolution values attained fromsimulations is presented in Figure 8a. Further, the result of an individual simulation is shown inFigure 8b. The results from this raytrace simulation are consistent with the results obtained whenassuming all errors are independent and add in quadrature – R ≈ (median value). So there-fore, if all errors contributed at their σ error limits, the OGRE spectrometer would not achieve itsspectral resolution goal of R > , but would still comfortably achieve its spectral resolutionrequirement of
R > at this wavelength.It should be noted that diffracted-order LSF resulting from a single simulation of the OGREspectrometer in Figure 8b demonstrates structure; it is apparent that two separate LSFs make upthis combined LSF. This structure suggest that the LSFs formed by the individual grating stackscould potentially be used to provide a higher resolution spectra, since the individual LSFs aremuch narrower than the combined LSF. Identification and subsequent analysis of these individualLSFs that make up the combined LSF would require thorough calibration of the instrument priorto launch to ensure all four LSFs on a single spectral detector – two from this grating module andanother two from a second grating module clocked 60 degrees relative to the first (geometry shownin Figure 2) – can be individiually identified. The potential for this analysis will be investigated inthe future. 20a) (b)
Figure 8
Results from 1,000 raytrace simulations of the OGRE spectrometer with misalignments, performance errors,and jitter at their σ limits (presented in Tables 1-4). (a): Distribution of the achieved spectral resolutions from the1,000 raytrace simulations. These values range from R ∼ ( σ low) to R ∼ ( σ high), with a medianspectral resolution of R = 1880 . (b): The diffracted-order LSF from a single raytrace simulation in (a). Structurewithin this LSF comes from the two misaligned grating stacks (two groups centered at x ∼ − . and x ∼ . inthe dispersion direction) and also from the individual gratings within each stack (quasi-horizontal lines within the twograting stack groups). This diffracted-order LSF achieves a spectral resolution of R ∼ . The previous simulations assumed all misalignments and performance errors contributed attheir σ error limit. In reality though, some misalignments will be well within their σ limit, whileothers might be closer to the periphery of their limits. To understand how this situation mightimprove the achievable resolution of the OGRE spectrometer, further raytrace simulations wereperformed. However, instead of assuming that all contributions to the LSF were at their σ lim-its, all misalignments were given random values within their σ parameter space. Performancecontributions (optic performance, grating resolution limit, and jitter) were kept at their σ lim-its, since values for these contributions are expected to be closer to their σ limits. Similar tothe previous simulations, this raytrace simulation was performed N sim = 1000 times with eachsimulation having unique misalignment values. The results of these simulations are presented inFigure 9. These simulations show that while there are still some incarnations of the spectrometerthat do not achieve the R > performance goal ( < . % of simulations), the vast majority ofspectrometers modelled achieved a spectral resolution beyond this goal ( > . %). In this manuscript, a comprehensive LSF error budget for the soft X-ray grating spectrometer on theOff-plane Grating Rocket Experiment (OGRE) was described. This error budget described poten-tial impacts to the LSF observed by the OGRE spectrometer including component misalignmentsand performance errors, derived realistic alignment tolerances for component misalignments, andelucidated how each misalignment and performance error impacted the observed LSF. The impactsfrom component misalignments and performance errors were combined first by assuming indepen-dent errors such that the errors added in quadrature. It was found that if all errors contribute at their21 igure 9
Results from 1,000 raytrace simulations of the OGRE spectrometer with misalignment values chosen ran-domly within their σ requirements, but with performance contributions (optic performance, grating period errors, andjitter) still contributing at their σ error limits. The achieved spectral resolutions range from R ∼ ( σ low) to R ∼ ( σ high), with a median spectral resolution of R = 2700 . σ error limits, the spectrometer would achieve a spectral resolution of ∼ − . This per-formance meets the OGRE spectral resolution requirement of R > comfortably, but doesnot meet the spectral resolution goal of
R > . However, if not all misalignments contributedat their σ error limits, but instead were randomly distributed within their ± σ error limits, thespectrometer meets the spectral resolution goal of R > for > . % of the simulations.These results suggest that the OGRE spectrometer should be able to comfortably meet its spectralresolution requirement of R > , and, depending on the exact values of misalignments duringassembly, even meet its spectral resolution goal of
R > . Disclosures
The authors have no relevant financial interests in the manuscript and no other potential conflictsof interest to disclose.
Acknowledgments
The work presented in this manuscript is supported by NASA grant NNX17AD19G, a NASASpace Technology Research Fellowship, and internal funding from The Pennsylvania State Uni-versity. The authors would like to thank current and past members of the McEntaffer researchgroup for their support of this project. This work makes use of PyXFocus, an open-source Python-based raytrace package.
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Benjamin D. Donovan is a PhD candidate at The Pennsylvania State University in the Departmentof Astronomy & Astrophysics. He received his B.S. in Physics & Astronomy from The Universityof Iowa in 2016, and his M.S. in Astronomy & Astrophysics from The Pennsylvania State Univer-sity in 2019. He is the recipient of a NASA Space Technology Research Fellowship. His currentresearch interests include the design and development of space-based astronomical instrumenta-tion. He is a member of SPIE.Biographies and photographs of the other authors are not available.23 ist of Figures
6, 7
Light is incident from point A onto the grating surface at a graze angle η and rel-ative to the groove direction by an angle Ψ . Equivalently, this incidence geometrycan be described in the spherical coordinate system by azimuth angle α and polarangle γ . Diffraction follows the generalized grating equation (Eq. 1) and light isdiffracted a distance L to azimuth angle β on the focal plane. The dispersion dis-tance between the n = 0 reflection and n -th diffraction order on this focal plane isgiven by x = nλL/d .2 Diffraction geometry for neighboring grating modules in the OGRE soft X-rayspectrometer. The optic assembly is divided into six 60° azimuthal sections. Be-hind each of these sections is a grating module containing 60 gratings. To max-imize diffraction efficiency, each grating module operates in the Littrow mountwhich requires α = β = δ . For the OGRE spectrometer, α = β = δ = 30 ° suchthat the neighboring grating modules diffract to the same location on the focalplane. This geometry is repeated two times to populate the entire 360° azimuthalspan of optic with grating modules. Depicted in the bottom-right is the coordinatesystem for this schematic view with +ˆz pointing into the page.3 Results from a raytrace simulation of the OGRE spectrometer showing the effectof a yaw misalignment of individual gratings in the grating module on the extentof the OGRE LSF in the dispersion (blue) and cross-dispersion (orange) direc-tions. The contribution in the dispersion direction is measured as a full-width athalf-maximum (FWHM), while the contribution in the cross-dispersion directionis 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 contributeto the LSF extent in the dispersion direction, but it has a noticeable impact on theLSF extent in the cross-dispersion direction.4 Left - A single reflection grating in the OGRE spectrometer. Each grating hasthree 2 mm wide ribs manufactured into the back of the substrate which allow lightto pass through the grating stack while still maintaining the precisely manufacturedwedge profile. Overlayed on this figure are the degrees of freedom (DoF) for eachreflection grating. The grooves for each grating converge in the − ˆz direction to thespectrometer focal plane. Right - An OGRE grating stack containing 30 individualreflection gratings stacked on top of one another.5 The anticipated alignment methodology for the two grating stacks to form a singlegrating module. The two stacks would be placed on the bottom surface, thenabutted against Pins A & B and either Pin C or Pin D. The precisely machined bot-tom plate constrains the ˆy, pitch, and roll of the grating stacks, while pins constrainthe remaining three degrees of freedom – ˆx, ˆz, and yaw. Here, Pins A & B con-strain the ˆx and yaw alignment and Pins C & D constrain the ˆz alignment betweengrating stacks. 24 CAD rendering of the OGRE payload showing the large separation (3.5 m) be-tween the forward assembly (comprised of the optic and grating assemblies) andthe detector assembly. Shown in the bottom-right corner is the coordinate systemreferenced in Section 2.3.7 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 mod-ules (geometry discussed in Figure 2), while the central detector samples the smallfraction 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 theleft-hand image). Shown in this image is the coordinate system for this particularspectral detector.8 Results from 1,000 raytrace simulations of the OGRE spectrometer with misalign-ments, performance errors, and jitter at their σ limits (presented in Tables 1-4).(a): Distribution of the achieved spectral resolutions from the 1,000 raytrace sim-ulations. These values range from R ∼ ( σ low) to R ∼ ( σ high),with a median spectral resolution of R = 1880 . (b): The diffracted-order LSFfrom a single raytrace simulation in (a). Structure within this LSF comes from thetwo misaligned grating stacks (two groups centered at x ∼ − . and x ∼ . in the dispersion direction) and also from the individual gratings within each stack(quasi-horizontal lines within the two grating stack groups). This diffracted-orderLSF achieves a spectral resolution of R ∼ .9 Results from 1,000 raytrace simulations of the OGRE spectrometer with misalign-ment values chosen randomly within their σ requirements, but with performancecontributions (optic performance, grating period errors, and jitter) still contributingat their σ error limits. The achieved spectral resolutions range from R ∼ ( σ low) to R ∼ ( σ high), with a median spectral resolution of R = 2700 . List of Tables ° azimuthal sec-tion of the OGRE optic assembly. Since the optic will be treated as an assembledunit, 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 di-rection (measured as a half-power diameter; HPD).2 Grating-induced errors to the observed LSF of the OGRE spectrometer, includingaberrations induced by the diffraction geometry, the finite resolution limit of the in-dividual gratings within a module, grating-level alignment, stack-level alignment,and module-to-optic alignment. Shown are the σ level (99.7%) requirements foreach error in all six degrees of freedom (DoF; if applicable) and the impact of theerror 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 thistable. 25 Errors induced by a misalignment of the forward assembly (aligned OGRE mirrormodule + OGRE grating module) into the observed LSF of the OGRE spectrome-ter. Shown are the σ level (99.7%) requirements for each error in all six degreesof freedom (DoF) and the impact of the error in both the dispersion direction (mea-sured 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 val-ues are reported for each error in this table.4 Errors introduced into the observed LSF of the OGRE spectrometer by a misalign-ment of a spectral detector relative to the nominal focal plane . Shown are the σσ