Limiting Spectral Resolution of a Reflection Grating Made via Electron-Beam Lithography
Casey T. DeRoo, Jared Termini, Fabien Grise, Randall L. McEntaffer, Benjamin D. Donovan, Chad Eichfeld
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Limiting Spectral Resolution of a Reflection Grating Made via Electron-Beam Lithography
Casey T. DeRoo, Jared Termini, Fabien Gris´e, Randall L. McEntaffer, Benjamin D. Donovan, andChad Eichfeld The University of Iowa,Dept. of Physics & Astronomy,Van Allen Hall, Iowa City, IA 52242, USA The Pennsylvania State University,Dept. of Astronomy & Astrophysics, 505 Davey Lab,University Park, PA 16802, USA The Pennsylvania State University,Materials Research Institute, N-153 Millennium Science Complex,University Park, PA 16802, USA (Received; Revised; Accepted)
Submitted to ApJABSTRACTGratings enable dispersive spectroscopy from the X-ray to the optical, and feature prominentlyin proposed flagships and SmallSats alike. The exacting performance requirements of these futuremissions necessitate assessing whether the present state-of-the-art in grating manufacture will limitspectrometer performance. In this work, we manufacture a 1.5 mm thick, 1000 nm period flat gratingusing electron-beam lithography (EBL), a promising lithographic technique for patterning gratingsfor future astronomical observatories. We assess the limiting spectral resolution of this grating byinterferometrically measuring the diffracted wavefronts produced in ± R ∼ Keywords: instrumentation: spectrographs, techniques: spectroscopic INTRODUCTIONEncoded in spectra are the physics of astronomical sources. Temperature, density, relative motion, velocity distri-butions, and ionization states can all be deduced with sufficiently detailed spectra. Gratings are a critical componentof dispersive spectrometers, which are employed across three orders of magnitude in wavelength space (1 – 1000 nm).The dispersive spectrometers of future observatories are tasked with addressing critical science questions requiringhigh spectral resolution R in order to untangle nearby line blends or detect faint features on top of a dominantbackground. For example, Lynx , an X-ray Strategic Mission Concept under study for the 2020 Decadal Survey onAstronomy and Astrophysics (Gaskin et al. (2019)), requires a dispersive grating spectrometer with
R > – 10 K) filamentary structures thought to host much of theUniverse’s baryonic material at the current epoch (Bregman et al. (2015) and references therein). Similarly, the LargeUltraviolet Optical Infrared Surveyor (LUVOIR), another Strategic Mission Concept, baselines two spectrometersoperating in the ultraviolet with resolutions ranging from R = 8,000 – 65,000 (LUMOS, France et al. (2017)) to R >
Corresponding author: Casey T. [email protected] a r X i v : . [ a s t r o - ph . I M ] S e p DeRoo et al. 2020 absorption spectroscopy, and characterizing the winds of metal-poor massive stars to understand their impact onfeedback processes at early cosmic epochs. For these missions, large-format ( (cid:38)
10 cm ) gratings are essential in orderto capture a large portion of the light from the large-aperture telescopes.The instrument optical designs for both Lynx and
LUVOIR benefit from grating customization. For example, intro-ducing a grating ‘chirp,’ where the period is intentionally varied across the grating, theoretically increases the spectralresolution of the transmission grating spectrometer concept for
Lynx (G¨unther & Heilmann (2019)). Furthermore,the POLLUX instrument, a UV-spectropolarimeter baselined for
LUVOIR , employs gratings patterned on freeformoptics i.e., optics with arbitrary deviations from a plane or curved surface (Muslimov et al. (2018a)). These custompatterns must be realized over areas (cid:38)
10 cm , and the resulting gratings blazed to offer high diffraction efficiency athigh dispersion.In addition to enabling instruments for these flagship concepts, customized gratings also permit novel small missionswith targeted science goals. Blazed gratings on curved surfaces would allow for further development of the two-element spectrometer concept for X-ray spectroscopy, a compact instrument that nonetheless would improve on the linedetection sensitivity of the Chandra and
XMM-Newton spectrometers by an order of magnitude (DeRoo et al. (2019)).By manipulating both the local groove density and groove direction of the grating pattern, aberration-correctinggratings can be realized (Beasley et al. (2019)). Aberration-correcting gratings enable unique diffraction geometries,such as point-to-point imaging with only one optical element for a fixed wavelength or a hyperspectral imager operatingfrom 400 – 1000 nm compact enough for a CubeSat format (Beasley et al. (2016)). Finally, customized gratings canimprove the performance of suborbital missions relying on efficiently blazed gratings (e.g., CHESS, Hoadley et al.(2016); OGRE, Donovan et al. (2019)).The gratings for all of these missions must be inherently capable of greater spectral resolution than needed for thescience case, as a realistic error budget for the spectrograph will degrade the resolution performance. Formally, thespectral resolution of a grating is limited by the total number of grooves G and working order n (Hutley (1982)), makingthe limiting spectral resolution of large-area gratings R = nG (cid:38) in principle. In practice, however, fabricationerrors dominate the achievable resolution. These fabrication errors are either related to the optical quality of thegrating substrate, such as slope errors or microroughness, or errors in the grating pattern itself, such as deviationsfrom the designed local grating period.The deviations from the designed local grating period are related to the groove placement accuracy of the techniqueused to pattern the grating – high groove placement accuracy yields a distribution of groove periods that closely matchthe grating’s design, while poor groove placement accuracy results in unwanted variation in the space between adjacentgrooves. These errors in groove placement enter as errors in the grating’s period in the generalized grating equation(Fig. 1): sin α + sin β = nλd sin γ , (1)as σ d , the error in the grating groove period d . For a given incidence geometry (set by α , the angle between thereflected beam and grating normal as projected into the focal plane, and γ , the cone angle between the center grooveand the incidence light), an error in d degrades the ability to measure the diffraction angle β and hence the ability todetermine the wavelength λ at a given order n . Thus, an error in d maps to an error in λ , limiting the resolution of aperfect spectrometer to R = λ/ ∆ λ ∼ d/σ d .In sum, a host of astrophysics missions benefit from gratings fabricated with high groove placement accuracy andcustomized for the instrument application. Hence, a high-accuracy patterning technique flexible enough to produce(1) blazed (2) grating patterns with variable line spacing (VLS) over (3) large areas on (4) freeform optical surfaces isdesirable. Several techniques promise a viable pathway towards meeting these requirements, including photolithographyon freeform substrates (Muslimov et al. (2018b)) and mechanical ruling using atomic force microscopy (Gleason et al.(2017)). In this paper, we focus on grating manufacture via electron-beam lithography (EBL). EBL is a precise,flexible lithography method which rasters a beam of energetic, collimated electrons across resist. EBL can createminute feature sizes ( ∼
10 nm, Manfrinato et al. (2013)), but has traditionally been limited to small areas ( (cid:46) ).However, recent advances in EBL patterning have permitted the production of a VLS grating with a nominal ∼ area (Miles et al. (2018)). EBL has also been used to write gratings with sculpted, triangulargroove profiles directly using greyscale lithography and thermal reflow (McCoy et al. (2018)). These direct-write blazedgratings have achieved high-diffraction efficiency in the X-ray when used in an echelle mounting (McCoy et al. (2020); imiting Spectral Resolution – EBL Reflection Grating Figure 1.
Cartoon of the generalized grating geometry. Light is incident on the grating at a cone angle γ , and reflects ordiffracts to a point on a semi-circle with the same cone angle. The undiffracted light forms an angle α with the grating normal,whereas diffracted light travels to an angle β . McCurdy et al. (2019)). Finally, EBL has been previously employed to a write grating on a curved optical surfacefor the
CRISM instrument onboard the
Mars Reconnaissance Orbiter , albeit with a large period (15.552 µ m) andover a ∼ area (Wilson et al. (2003)). Thus, EBL has demonstrated large-formats, blazing, curved substratepatterning, and VLS pattering, making the method a viable technical path towards realizing gratings suitable forfuture astronomical instruments.However, an assessment of the groove placement accuracy afforded by EBL for astronomical gratings is lacking.Measurements of the spectral resolving power of X-ray spectrometer systems under test offer lower limits on theperiod error (e.g., DeRoo et al. (2020); Donovan et al. (2020); Heilmann et al. (2019). However, these studies offerno spatial information about the achieved grating periodicity, and are routinely limited by the inherent width ofthe fluoresced X-ray lines or the focus quality of the employed telescope. Measuring the period error of EUV/X-raygratings for use in synchrotrons has been previously done (e.g., Voronov et al. (2017); Gleason et al. (2017)); however,these gratings are on thick substrates not suitable for astronomical use given the constraints on packing geometry andinstrument mass.In this paper, we characterize the groove placement accuracy achieved on an EBL-patterned astronomical gratingusing an optical interferometer. Interferometric measurements offer information about the frequency content andspatial distribution of period errors and are assessed independently of a spectrometer system. Moreover, the gratingin the present study is patterned using the same tooling used to produce the large-format VLS grating of Miles et al.(2018) and the direct-write blazed gratings of McCurdy et al. (2019). Thus, these measurements characterize thepresent state-of-the-art for making high-resolution, customized gratings with EBL. A description of the grating undertest and the interferometric measurements conducted are described in Section 2. In Section 3, the groove placementaccuracy and the derived groove period error are presented, along with a method of verifying the internal consistencyof our measurements and an assessment of the noise inherent in our interferometric measurement. Finally, a discussionof the implications of these results for astrophysical instruments and a description of an interferometric technique forassessing EBL patterns on curved substrates is given in Section 4. EXPERIMENTAL METHOD2.1.
Measuring Gratings Interferometrically
Both the figure of a planar, constant-period grating and the optical quality of the diffracted orders can be assessedinterferometrically, provided the constraints can be satisfied (Hutley (1982)). Measuring the grating’s figure is straight-forward – in reflection, the grating behaves like a mirror. Thus, the optical figure of the grating can be measured withan interferometer equipped with a transmission flat by placing the grating at normal incidence to the beam (Fig. 2A).A commercial interferometer integrates over the produced fringe patterns to yield a height map over the surface ofthe grating. In this paper, this measurement of the grating figure is referred to as the 0th order height map H ( x, y ),where x is the coordinate in the dispersion direction and y is along the groove direction.To measure the wavefront produced by the diffracted orders, the grating is placed into a geometry in which adiffracted order propagates back along the incident beam. This is achieved by setting γ = 90 ◦ such that diffractionhappens entirely in the plane of incidence, and rotating the grating about the groove direction such that the condition DeRoo et al. 2020 sin α = nλ d , (2)is realized. This condition stems from evaluating Eq. 1 when β = α . In this geometry, a coherent, back-diffractedwavefront produced by the grating can be measured by the interferometer. As a result of projection effects, thiswavefront is diminished in lateral extent by a factor of cos α relative to the 0th order height map. The height mapfor diffracted order n is referred to as H n ( x cos α n , y ), where α n is the angle α for which Eq. 2 is satisfied for a givenorder n . Figure 2.
Geometry for the interferometric grating measurement. (A) The incidence geometry for measuring the 0th order ofthe grating. In this geometry, the grating behaves like a mirror and the figure of the optic is measured directly. (B) Rotationabout the groove direction (out of the page) such that Eq. 2 is satisfied aligns +1st order with the incidence wavefront,permitting the interferometer measurement of the back-diffracted wavefront. (C) Similar to (B), the opposite rotation permitsthe measurement of -1st order interferometrically.
An ideal grating will produce a back-diffracted wavefront that is perfectly in-phase – the phase difference across thegrating from the non-zero incidence angle are exactly offset by the phase difference induced by the grating pattern.Phase errors in the diffracted wavefront hence have one of two sources: (1) the figure error of the grating viewed inprojection results in path length errors and thus phase errors between different physical areas of the grating or (2)local deviations from the average period change the local angle of diffraction, resulting in a difference in path length(and hence phase) when traversing the interferometric cavity. For a real (i.e., non-idealized) grating, both of these imiting Spectral Resolution – EBL Reflection Grating H ( x, y ). At small incidence angles, these errors are sym-metric about order i.e., identical for back-diffracted wavefronts with the same value of | n | . In the latter case of perioderror, grooves offset from their position in an idealized grating fail to cancel the phase difference introduced by placingthe grating at a non-zero incidence angle. We define this offset in idealized groove position as ∆( x, y ) for a given tracein the dispersion direction. ∆( x, y ) quantifies the groove placement accuracy of a grating patterning technique (seeSec. 1), as it measures the offset between the realized and ideal groove positions. The phase errors introduced due to∆( x, y ) are antisymmetric about order since grooves are offset closer to or farther away from the source wavefront atopposite incidence angles, producing the opposite phase shift for ± n .This difference in symmetry can be exploited to isolate these sources of error, yielding either the groove offset ∆( x, y )or the figure error H ( x, y ) from diffracted measurements of opposite orders H ± n ( x, y ). The height map measured bya commercial interferometer is related to the spatially-varying phase map φ ( x, y ) produced by the grating: φ ( x, y ) = 4 πλ H n ( x, y ) , (3) modulo a constant overall phase offset. Following the definitions of Gleason et al. (2017) and using Eq. 3, it can beshown that: H ( x, y ) = H + n ( x cos α n , y ) + H − n ( x cos α n , y )2 cos α n , (4)∆( x, y ) = dλ (cid:0) H + n ( x cos α n , y ) − H − n ( x cos α n , y ) (cid:1) , (5)where d is the average period of the grating under measurement and λ the operating wavelength of the interferometer.Next, we relate the measured groove offset ∆( x, y ) to period error as a function of position on the grating σ d ( x, y )by considering the origin of a groove offset in one dimension. We define the distance between two grating grooves ( i,j ) in the dispersion direction as x i,j , which can be written as: x i,j = ( j − i ) d + ∆( x i,j ) (6)where d is the average period of the grating, ( j − i ) is the total number of grooves separating the two under examinationand ∆( x i,j ) encompasses any remaining offset i.e., any deviation from an ideal grating. ∆( x i,j ), in turn, is the sum ofthe period errors for all the grooves between the i th and j th (see Fig. 3):∆( x i,j ) = j (cid:88) k = i σ k . (7)where σ k is the error in the period of the k th groove.To calculate a localized average period error σ d ( x ) as a function of position on the grating, we next make twoassumptions. First, we assume that the spatial sampling in the dispersion direction dx constitutes many periods i.e.( j − i ) >> dx/d >>
1. Next, we assume that the period error for each groove is small relative to the overallperiod of the grating, σ k << d , such that dx can be faithfully approximated as ( j − i ) d . Under these assumptions,Eq. 7 can be converted to an integral and differentiated to yield: σ d ( x ) = d d ∆( x ) dx (8)where σ d ( x ) represents the localized average period error. Eq. 8 has a straightforward interpretation – since the errorin groove position accumulates at a rate of σ d ( x ) per groove, the error d ∆( x ) over a spatial interval dx is simply thenumber of grooves contained in the interval ( dx/d ) multiplied by σ d ( x ).This expression is easily generalized to two dimensions by considering that the groove offset ∆( x, y ) relative to anarbitrary reference point is determined wholly by the average period error accumulating in the dispersion direction x . Eq. 5 and Eq. 8 thus provide a means to move from a set of interferometric measuremements of diffracted orders H ± n ( x cos α n , y ) to a map of the average period error over the surface of the grating σ d ( x, y ). DeRoo et al. 2020
Figure 3.
A diagram depicting the differences between an ideal grating (black, dashed line) and the as-manufactured grating(red, solid line). The distance between the i th and j th facet is equal to the number of periods between them ( j − i ) d plus agroove offset ∆( x ). This groove offset is in turn related to the average period error by Eqs. 7 and 8. EBL-Written Grating Measurements
To assess the groove placement accuracy of astronomical gratings made using EBL, we manufactured a 50 mm ×
50 mm, 1000 nm period grating with laminar grooves (Fig. 4). The grating substrate is a 6 inch diameter, 1.5 mmthick silicon wafer. A 30 nm silicon nitride layer was deposited onto the silicon wafer with low-pressure chemical vapordeposition and served as a hard mask for the transfer of the pattern. This wafer was next coated with ZEP520A (1:1dilution in anisole) and patterned at The Pennsylvania State University’s Material Research Institute using a RaithEBPG5200 EBL tool. Following development, the pattern was transferred into the silicon substrate using a plasmaetch performed by a Plasma-therm Versalock tool. The process for manufacturing astronomical gratings via EBL isdescribed in greater detail in Miles et al. (2018).
Figure 4.
The 50 mm ×
50 mm laminar grating written using EBL measured in this study. The grating area is visible in thelower center of the 6 in. diameter Si wafer as a darker, iridescent square. The grating substrate is held in a three-point mount,which is in turn affixed to rotation stages permitting alignment to the interferometer.
For the present work, we measure both the grating’s figure (0th order) and wavefront of the +1st and -1st diffractedorders in back-diffraction. With respect to Eq. 2, the back-diffaction angle α ± for these measurements is 18.44 ◦ .Measurements were taken with a 4D Technologies AccuFizH100S Fizeau interferometer operated in dynamic modein order to minimize the impact of environmental vibration during integrations (Brock et al. (2005)). This tool isequipped with a 6MP camera for ultra-fine sampling; the effective pixel scale of measurements operated in this modeis 0.043 mm/pix in 0th order. imiting Spectral Resolution – EBL Reflection Grating Figure 5.
The ITF of the AccuFizH100S Fizeau interferometer employed to measure the EBL grating produced for this study.The measured ITF is shown as a solid line, while the best-fit to the ITF is shown as dashed line.
The measurement error of the wavefronts is estimated using the repeatability of the interferometer. The gratingfigure (0th order) was measured in separate instances, between which the grating was realigned to the interferometer.We assume that this repeatability error is comparable to the error present in our measurements at separate orders, aseach order also requires a distinct alignment to the interferometer beam. This error is used to assess the error presentin our calculation of 0th order from ± RESULTSThe measured height maps of 0th, +1st, and -1st orders ( H ( x, y ) , H +1 ( x, y ) , H − ( x, y ) respectively) are shown inFig. 6. These height maps are normalized such that the average value across the height map is zero (i.e., that pistonis zero) so they can be compared directly. These three measurements, along with the interferometer repeatability usedas a proxy for the error implicit in these height maps, form the basis for the analysis presented in Sections 3 and 4.3.1. Methodology Verification – Predicting Figure from the Diffracted Wavefronts
As a cross-check on the self-consistency of the height map measurements, we use the height maps of the +1st and-1st orders to predict the figure of the grating as measured by the 0th order height map. From Eq. 4, we see that: H +1 ( xα ± , y ) + H − ( x cos α ± , y )2 cos α ± − H ( x, y ) = 0 , (9)where zero on the right hand side of the expression is interpreted in a statistical sense i.e., is consistent within theerror of the interferometer. The left side of Fig. 7 displays the result of evaluating the left hand side of Eq. 9 with themeasured data. We find that this difference map is centered about zero as expected. Moreover, we find a characteristicvariation of 7.0 nm RMS across the grating surface, consistent with the variation expected due to interferometermeasurement noise (9.8 nm RMS). Thus, we conclude that our measurements are consistent with Eq. 9, bolsteringconfidence in the application of the methodology outlined in Section 2.1. DeRoo et al. 2020
Figure 6.
The measured height maps of 0th order (left), +1st order (middle), and -1st order (right). All height maps are inmicrons, have the same color scale, and are plotted over the 50 mm ×
50 mm extent of the grating.
Figure 7. (A) The difference between the predicted grating figure (0th order) as calculated from H +1 ( x, y ) and H − ( x, y ) andthe measured 0th order. This height map is equivalent to evaluating the lefthand side of Eq. 9. (B) The error in the differencemap, as calculated by propagating the interferometer repeatability via Eq. 9. By either a peak-to-valley (PV) or RMS metric,the difference map agrees with zero to within measurement error. Groove Displacement and Groove Period Error
We next compute the position-dependent groove displacement ∆( x, y ) from the height maps of the diffracted ordersusing Eq. 5. The result is shown in Fig. 8. The left side of Fig. 8 is the displacement map over the wholegrating surface, ∆( x, y ), while the right side shows several representative traces of groove displacement across thedispersion direction (i.e., ∆( x, y = c ), where c is a chosen constant). Note that we have remapped the pixel scale ofthese ± α ± reduction in spatial sampling. In doing so, they can becompared directly to the 0th order height map (Fig. 6), albeit with coarser spatial sampling (0.045 mm/pixel vs. 0.043mm/pixel). Periodic structure in this displacement map on the scale of a couple of millimeters is evident; an analysisof the power spectral density (PSD) of this measurement is performed in Sec. 3.3.From the groove displacement map, the average period error as a function of position is derived using Eq. 8.Similar to Fig. 8, the left side Fig. 9 shows the average period error over the grating surface, while, on the right,the average period errors at fixed positions along the groove dimension are shown. To examine the distribution of theaverage period error over the entire grating, we construct a histogram of the average period error from the 1.2 × pixels sampling the grating (Fig. 10). We assume Gehrels errors in each bin (Gehrels (1986)), and fit the resulting imiting Spectral Resolution – EBL Reflection Grating Figure 8. (A) A map of the groove displacement ∆( x, y ) over the grating surface as calculated from H +1 ( x, y ) and H − ( x, y )via Eq. 5. (B) Traces across the map of ∆( x, y ) taken at fixed y positions. distribution with lmfit , a Python-based curve-fitting package (Newville et al. 2019). We find that the groove perioderrors are satisfactorily described ( χ r = 0.95) by a Gaussian distribution. Figure 9. (A) The average period error across the grating surface as calculated from Fig. 8 and Eq. 8. (B) Traces across theaverage period error map at fixed y positions. We calculate the RMS of the raw average period error map shown in Fig. 9A to be 0.028 nm. This establishesthe correct order of a typical period error over the EBL-written grating; however, it should be both corrected for theimpact of the ITF and contextualized relative to the interferometer’s measurement error in order to provide a faithfulassessment of the groove placement accuracy of EBL.3.3.
Frequency Content of Average Period Error
While the groove period errors are found to be distributed as a uniform Gaussian in magnitude, there is a clearspatial correlation present in Fig. 9A. To elucidate this spatial correlation, we compute the power spectral density(PSD) of the period error map in the dispersion direction by averaging the one-dimensional PSDs of each row i.e., https://lmfit.github.io/lmfit-py/index.html DeRoo et al. 2020
Figure 10.
A histogram of the average period errors in Fig. 9A. A Gaussian fit to this distribution is shown as a red dashedline. at a fixed groove direction position. As a cross-check on our representative PSD, we calculate the integral of thisrepresentative PSD over frequency and find that it agrees with the standard deviation of the groove period error mapto within 5% error, as expected via Parseval’s theorem.We next correct this representative PSD with the best-fit ITF from the interferometer calibration (Fig. 5, red line);this correction results in a <
5% change in σ (as calculated by integrating the PSD) relative to the uncorrected PSD.The ITF-corrected representative PSD is shown as a black line in Fig. 11. The average period error σ d as calculatedfrom this PSD is 0.029 nm. Contributing to this average period error is two primary peaks at 0.44 ± ± H ± and propagating this error through Eqs. 5 and 8. The resultingPSD from the estimated error is displayed as a red dashed line in Fig. 11. Note that we do not correct this error PSDwith the ITF, as this is an estimate of the random error inherent in the measurement as opposed to the systematicunderprediction of power from the imperfect ITF. We use the error PSD to estimate the ultimate sensitivity of theinterferometric measurements described in Sec. 2.1 by integrating this PSD to calculate a limiting average period error σ d,noise . Using the interferometer noise floor in this way results in σ d,noise of 0.028 nm. DISCUSSION4.1.
Implications for Astronomical Spectroscopy Missions
The average groove period error calculated in Sec. 3 has implications for the adoption of EBL-written gratings byfuture astronomical spectroscopy missions. Most science applications for grating spectroscopy, such as line detectionor assessing velocity fields, have a figure of merit dependent on the spectral resolution R of an instrument, where R = λ/ ∆ λ and ∆ λ is the full width half maximum (FWHM) of the wavelength uncertainty. Following this useof FWHM in the definition of R , we adopt ∆ d = 2 . σ d = 0.068 nm. The factor of 2.35 is derived by relating σ and FWHM for a Gaussian; we assert that the assumption of this relationship is well-justified given the Gaussiandistribution of period errors shown in Fig. 10.The resulting estimate of R for the EBL-written grating tested here is R = d/ ∆ d = 14,600. However, there aretwo important caveats to this resolution estimate. First, we note that this resolution estimate is dependent on theassumed period d . In general, the period of a grating can be assessed in two ways: (1) the average feature size canbe measured with nanometer-scale metrology, such as an AFM or SEM, or (2) under illumination by a pencil beam ofknown wavelength, the relative angle between diffracted and reflected light can be measured precisely and d calculatedby use of Eq. 1. However, in both instances, the grating period is assessed locally i.e., in a small area as comparedto the entirety of the grating. Hence, this period may not be representative of the average period over the entirety ofthe grating, as assumed for the calculation of the average groove period error in Sec. 3.2. However, recognizing thatthe overall dependence of R on d is merely d − , we note that a percent uncertainty in d directly maps to a percentuncertainty in R . We thus address this quandary of not knowing the true value of d recognizing that, as most often d is known to a few percent, the R reported here should also not be taken to be more accurate than a few percent. imiting Spectral Resolution – EBL Reflection Grating Figure 11.
The PSD of the average groove period error (Fig. 9A, solid black line) and of the error in the average grooveperiod error (dashed red line), as estimated by propagating the interferometer repeatability via Eqs. 5 and 8.
The second, arguably more important, caveat is the issue of the uncertainty inherent in the interferometric measure-ment. The reported average period error contains contributions from both the true period error of the grating andthe interferometer noise floor. As illustrated in Fig. 11, the PSD of the interferometric measurement uncertainty iscomparable to the PSD of the average period error itself. We therefore argue that the contribution of the interferometernoise floor dominates over the true period error of the grating, and that the estimated limiting spectral resolution R =14,600 of this grating should be treated as a lower bound for comparable EBL-written gratings.In a mission context, period error should be handled as only one component of a comprehensive error budget.Nonetheless, our findings indicate that EBL-written gratings are suitable for missions such as Arcus and
Lynx giventhe target spectral resolutions of
R >
R >
EBL Error Contributions at Specific Frequencies
Only two frequency components, 0.44 ± ± µ m. While the measured spatial scale does not correspond directly to the scale of this writefield to within 3 σ error (4.93 ± DeRoo et al. 2020 dependent on the interferometer pixel scale calibrated at the time of measurement. A systematic error of 1.2% in thispixel calibration scale is sufficient to explain the difference between the as-measured frequency components and theexpected frequencies based on the EBL write, and is a reasonable magnitude given the calibration process.Similarly, the low frequency component is in keeping with the frequency of a calibration step performed during theEBL-writing process. For this particular write, the EBL tool alignment is checked every 7 minutes during the course ofthe write. During this alignment check, the EBL tool translates to fixed alignment markers patterned on the substrateoutside of the grating area and recalibrates its position based on these markers. Examining the tool log, we find thatthe average separation in the dispersion direction between these calibration points is 2.34 ± σ d,f = 0.009 nm to the overall power. The high frequency peak contributes σ d,f = 0.008nm to the total measured power, as estimated by integrating the PSD over the range of 4.8 - 5.0 cycles/mm. Addingthese in quadrature, a total of σ d,EBL = (cid:113) σ d,f + σ d,f = 0.012 nm of the measured power is directly attributable tothe EBL-writing process itself. We note that no effort has yet been made to minimize the impact of these calibrationsteps during the EBL writing process, and hence it may be possible to reduce their impact on σ d .Adopting σ d,EBL as a best-estimate of the groove period error inherent in the EBL writing process yields a limitingspectral resolution of R ∼ Future Measurements of Curved Substrates
An important open question is whether the EBL groove placement accuracy as quantified here will translate faithfullyto the patterning of more complex grating patterns, e.g., gratings with arbitrary groove orientations or freeformsurfaces. Such gratings break the simplifying symmetry of the interferometric measurement technique for constantperiod, flat gratings as presented in Sec. 2.1. In principle, however, the interferometric technique can be adapted toassess the expected groove placement accuracy on a customized EBL grating by writing a grating pattern yielding aback-diffracted wavefront when illuminated by a plane wave.To perform such a measurement, a test substrate with a figure representative of the desired customized grating isneeded. Assuming a fixed incidence geometry on this test substrate, the local groove density and orientation yieldinga back-diffracting wavefront can be calculated numerically via a raytracing program. This back-diffracting gratingpattern would then be written on a test substrate, and the back-diffracting wavefront measured interferometrically.Additionally, detailed knowledge of the figure of the test substrate is required, either from manufacturing metrologyor via measurement with a computer-generated hologram. This independent measurement of the test substrate can beused to account for the phase errors introduced in a back-diffracted wavefront due to figure. Subtracting the impact ofthe test substrate figure from the back-diffraction interferogram yields a residual phase error due to imperfect grooveplacement on this curved substrate.As a concrete example, we have calculated the customized grating pattern required to assess the groove placementaccuracy on a spherical grating. The assumed substrate and incidence geometry for this test grating is shown in Fig.12A. The sphere is assumed to have a diameter of 76.2 mm and a radius of curvature of 220 mm. The sphere wouldbe patterned with a grating over at least the central 50 mm square and illuminated by a plane wave of the same size.The groove direction vector and grating period required to yield a back-diffracted wavefront assuming this substrateand incidence geometry are shown as the lefthand and righthand plots in Fig. 12B. Assessing the groove placementaccuracy of EBL on a curved substrate such as this simple sphere would serve as a proof-of-concept for the customizedgratings discussed in Sec. 1, and reduce the risk of adoption for instrument concepts featuring these gratings such asthe two-element spectrometer (DeRoo et al. (2019)). imiting Spectral Resolution – EBL Reflection Grating Figure 12. (A) A diagram showing the optical bench configuration for assessing the groove displacement of a spherical EBLgrating. (B) The grating groove orientation (left) and period map (right) required to yield a back-diffracting sphere. Errors inrealizing this pattern with EBL lithography will result in phase errors in the interferogram of the back-diffracted order.5.
CONCLUSIONSWe have fabricated a large-format, 1000 nm period grating on a relatively thin (1.5mm) Si substrate, and assessedits limiting spectral resolution in the context of astronomical spectroscopy missions. Our core findings are as follows: • The groove placement accuracy and average period error σ d over the grating surface have been calculated frominterferometric measurements of the ± σ d = 0.029 nm. • Using only the ± • We find that the measured distribution of period errors on this EBL-written grating to be well-described by aGaussian distribution. • Based on a representative PSD, we identify two frequency components associated with the stitching error of theEBL fabrication process. These frequency components contribute σ d,f = 0.009 nm and σ d,f = 0.008 nm to theoverall power of the average period error. • Outside of these two EBL-related frequency components, the PSD is similar in shape and magnitude to the PSDexpected from propagating the interferometer measurement error. Moreover, the overall power attributable tothe error is 0.028 nm. • We estimate the limiting spectral resolution of this EBL-written grating as R ∼ d/ ∆ d , where ∆ d is the FWHM ofthe average period error. This yields R = 14,600 for EBL-written gratings. We argue that, given the contributedpower of the interferometric measurement error, this should be interpreted as a lower bound. Performing thesame assessment for the stitching error features with power above the interferometer noise floor yields R =35,000.Based on these measurements, we conclude that, in principle, EBL-written gratings are a suitable grating technologyfor spectroscopy missions requiring periods ∼ R >
DeRoo et al. 2020 supporting missions of
R >
R > σ d,noise scales with d and is of order a few tenths of an Angstromfor a grating with period d = 1000 nm, improvements to interferometric measurement techniques may also be neededto support the testing and calibration of gratings for systems R >
PyXFocus , an open source Python-based raytracingpackage. REFERENCES
Beasley, M., Goldberg, H., Voorhees, C., & Illsley, P. 2016,in CubeSats and NanoSats for Remote Sensing, ed. T. S.Pagano, Vol. 9978, International Society for Optics andPhotonics (SPIE), 66 – 73, doi: 10.1117/12.2238206Beasley, M., McEntaffer, R., & Cunningham, N. 2019, inUV, X-Ray, and Gamma-Ray Space Instrumentation forAstronomy XXI, ed. O. H. Siegmund, Vol. 11118,International Society for Optics and Photonics (SPIE),351 – 359, doi: 10.1117/12.2528758Bouret, J.-C., Neiner, C., de Castro, A. I. G., et al. 2018, inSpace Telescopes and Instrumentation 2018: Ultravioletto Gamma Ray, ed. J.-W. A. den Herder, S. Nikzad, &K. Nakazawa, Vol. 10699, International Society forOptics and Photonics (SPIE), 851 – 861,doi: 10.1117/12.2312621Bregman, J. N., Alves, G. C., Miller, M. J., &Hodges-Kluck, E. 2015, Journal of AstronomicalTelescopes, Instruments, and Systems, 1, 045003Brock, N., Hayes, J., Kimbrough, B., et al. 2005, in NovelOptical Systems Design and Optimization VIII, ed. J. M.Sasian, R. J. Koshel, & R. C. Juergens, Vol. 5875,International Society for Optics and Photonics (SPIE),101 – 110, doi: 10.1117/12.621245de Groot, P., & de Lega, X. C. 2006, in Fringe 2005, ed.W. Osten (Berlin, Heidelberg: Springer BerlinHeidelberg), 30–37DeRoo, C. T., McEntaffer, R. L., Donovan, B. D., et al.2019, in Optics for EUV, X-Ray, and Gamma-RayAstronomy IX, ed. S. L. O’Dell & G. Pareschi, Vol.11119, International Society for Optics and Photonics(SPIE), 287 – 295, doi: 10.1117/12.2528817 DeRoo, C. T., McEntaffer, R. L., Donovan, B. D., et al.2020, The Astrophysical Journal, 897, 92,doi: 10.3847/1538-4357/ab9a41Donovan, B. D., McEntaffer, R. L., Tutt, J. H., et al. 2019,in Optics for EUV, X-Ray, and Gamma-Ray AstronomyIX, ed. S. L. O’Dell & G. Pareschi, Vol. 11119,International Society for Optics and Photonics (SPIE),296 – 309, doi: 10.1117/12.2528858Donovan, B. D., McEntaffer, R. L., DeRoo, C. T., et al.2020, Journal of Astronomical Instrumentation,SubmittedFrance, K., Fleming, B., West, G., et al. 2017, in UV,X-Ray, and Gamma-Ray Space Instrumentation forAstronomy XX, ed. O. H. Siegmund, Vol. 10397,International Society for Optics and Photonics (SPIE),303 – 324, doi: 10.1117/12.2272025Gaskin, J. A., Swartz, D., Vikhlinin, A. A., et al. 2019,Journal of Astronomical Telescopes, Instruments, andSystems, 5, 021001Gehrels, N. 1986, Astrophysical Journal, 303, 336Gleason, S., Manton, J., Sheung, J., et al. 2017, inAdvances in Metrology for X-Ray and EUV Optics VII,Vol. 10385, 1038506Goodman, J. W. 1996, Introduction to Fourier Optics, 2ndedn. (New York City, New York: McGraw-Hill)G¨unther, H. M., & Heilmann, R. K. 2019, Journal ofAstronomical Telescopes, Instruments, and Systems, 5, 1, doi: 10.1117/1.JATIS.5.2.021003Heilmann, R. K., Kolodziejczak, J., Bruccoleri, A. R.,Gaskin, J. A., & Schattenburg, M. L. 2019, AppliedOptics, 58, 1223 imiting Spectral Resolution – EBL Reflection Grating15