Limiting Spectral Resolution of a Reflection Grating Made via Electron-beam Lithography

Both the figure of a planar, constant-period grating and the optical quality of the diffracted orders can be assessed interferometrically, provided the constraints can be satisfied (Hutley 1982). Measuring the grating's figure is straightforward—in reflection, the grating behaves like a mirror. Thus, the optical figure of the grating can be measured with an interferometer equipped with a transmission flat by placing the grating at normal incidence to the beam (Figure 2(A)). A commercial interferometer integrates over the produced fringe patterns to yield a height map over the surface of the grating. In this paper, this measurement of the grating figure is referred to as the zeroth order height map ${{ \mathcal H }}_{0}(x,y)$, where x is the coordinate in the dispersion direction and y is along the groove direction.

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To measure the wave front produced by the diffracted orders, the grating is placed into a geometry in which a diffracted order propagates back along the incident beam. This is achieved by setting γ = 90° such that diffraction happens entirely in the plane of incidence, and rotating the grating about the groove direction such that the condition

$$\begin{eqnarray}&&\sin \alpha =\displaystyle \frac{n\lambda }{2d},\end{eqnarray} \tag{ 2 }$$

is realized. This condition stems from evaluating Equation (1) when β = α. In this geometry, a coherent, back-diffracted wave front produced by the grating can be measured by the interferometer. As a result of projection effects, this wave front is diminished in lateral extent by a factor of $\cos \alpha$ relative to the zeroth order height map. The height map for diffracted order n is referred to as ${{ \mathcal H }}_{n}(x\cos {\alpha }_{n},y)$, where ${\alpha }_{n}$ is the angle α for which Equation (2) is satisfied for a given order n.

An ideal grating will produce a back-diffracted wave front that is perfectly in-phase—the phase difference across the grating from the nonzero incidence angle are exactly offset by the phase difference induced by the grating pattern. Phase errors in the diffracted wave front hence have one of two sources: (1) the figure error of the grating viewed in projection 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., nonidealized) grating, both of these sources of error are present simultaneously. These errors are discussed in greater detail in Gleason et al. (2017); we summarize key aspects of their work as pertains to interferometric grating measurement in the following paragraphs.

For grating figure, phase differences are proportional to ${{ \mathcal H }}_{0}(x,y)$. At small incidence angles, these errors are symmetric about order i.e., identical for back-diffracted wavefronts with the same value of $| n|$. In the latter case of period error, grooves offset from their position in an idealized grating fail to cancel the phase difference introduced by placing the grating at a nonzero incidence angle. We define this offset in idealized groove position as ${\rm{\Delta }}(x,y)$ for a given trace in the dispersion direction. Δ(x, y) quantifies the groove placement accuracy of a grating patterning technique (see Section 1), as it measures the offset between the realized and ideal groove positions. The phase errors introduced due to Δ(x, y) are antisymmetric about order since grooves are offset closer to or farther away from the source wave front at opposite 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 Δ(x, y) or the figure error ${{ \mathcal H }}_{0}(x,y)$ from diffracted measurements of opposite orders ${{ \mathcal H }}_{\pm n}(x,y)$. The height map measured by a commercial interferometer is related to the spatially varying phase map ϕ(x, y) produced by the grating:

$$\begin{eqnarray}&&\phi (x,y)=\displaystyle \frac{4\pi }{\lambda }{{ \mathcal H }}_{n}(x,y),\end{eqnarray} \tag{ 3 }$$

modulo a constant overall phase offset. Following the definitions of Gleason et al. (2017) and using Equation (3), it can be shown that:

$$\begin{eqnarray}&&{{ \mathcal H }}_{0}(x,y)=\displaystyle \frac{{{ \mathcal H }}_{+n}(x\cos {\alpha }_{n},y)+{{ \mathcal H }}_{-n}(x\cos {\alpha }_{n},y)}{2\cos {\alpha }_{n}},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}(x,y)=\displaystyle \frac{d}{\lambda }\left({{ \mathcal H }}_{+n}(x\cos {\alpha }_{n},y)-{{ \mathcal H }}_{-n}(x\cos {\alpha }_{n},y)\right),\end{eqnarray} \tag{ 5 }$$

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 Δ(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:

$$\begin{eqnarray}&&{x}_{i,j}=(j-i)d+{\rm{\Delta }}({x}_{i,j}),\end{eqnarray} \tag{ 6 }$$

where d is the average period of the grating, (j − i) is the total number of grooves separating the two under examination and Δ(x_{i, j}) encompasses any remaining offset, i.e., any deviation from an ideal grating. Δ(x_{i, j}), in turn, is the sum of the period errors for all the grooves between the ith and jth (see Figure 3):

$$\begin{eqnarray}&&{\rm{\Delta }}({x}_{i,j})=\displaystyle \sum _{k=i}^{j}{\sigma }_{k},\end{eqnarray} \tag{ 7 }$$

where σ_k is the error in the period of the kth groove.

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To calculate a localized average period error σ_d(x) as a function of position on the grating, we next make two assumptions. First, we assume that the spatial sampling in the dispersion direction dx constitutes many periods, i.e., $(j-i)\gg 1$ such that ${dx}/d\gg 1$. Next, we assume that the period error for each groove is small relative to the overall period of the grating, ${\sigma }_{k}\ll d$, such that dx can be faithfully approximated as (j − i)d. Under these assumptions, Equation (7) can be converted to an integral and differentiated to yield:

$$\begin{eqnarray}&&{\sigma }_{d}(x)=d\displaystyle \frac{d{\rm{\Delta }}(x)}{{dx}},\end{eqnarray} \tag{ 8 }$$

where σ_d(x) represents the localized average period error. Equation (8) has a straightforward interpretation–since the error in groove position accumulates at a rate of σ_d(x) per groove, the error dΔ(x) over a spatial interval dx is simply the number 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 Δ(x, y) relative to an arbitrary reference point is determined wholly by the average period error accumulating in the dispersion direction x. Equations (5) and (8) thus provide a means to move from a set of interferometric measurements of diffracted orders ${{ \mathcal H }}_{\pm n}(x\cos {\alpha }_{n},y)$ to a map of the average period error over the surface of the grating σ_d(x, y).

To assess the groove placement accuracy of astronomical gratings made using EBL, we manufactured a 50 mm × 50 mm, 1000 nm period (i.e., 1000 gr mm⁻¹) grating with laminar grooves (Figure 4). The grating substrate is a 6 inch diameter, 1.5 mm thick silicon wafer. A 30 nm silicon nitride layer was deposited onto the silicon wafer with low-pressure chemical vapor deposition and served as a hard mask for the transfer of the pattern. This wafer was next coated with ZEP520A (1:1 dilution in anisole) and patterned at The Pennsylvania State University's Material Research Institute using a Raith EBPG5200 EBL tool. Following development, the pattern was transferred into the silicon substrate using a plasma etch performed by a Plasma-therm Versalock tool. The process for manufacturing astronomical gratings via EBL is described in greater detail in Miles et al. (2018).

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For the present work, we measure both the grating's figure (zeroth order) and wave front of the +first and −first diffracted orders in back-diffraction. With respect to Equation (2), the back-diffraction angle α_±1 for these measurements is 1844. Measurements were taken with a 4D Technologies AccuFizH100S Fizeau interferometer (operating wavelength λ = 632.8 nm). The AccuFizH100S was operated in dynamic mode in order to minimize the impact of environmental vibration during integrations (Brock et al. 2005). This tool is equipped with a 6MP camera for ultra-fine sampling; the effective pixel scale of measurements operated in this mode is 0.043 mm pix⁻¹ in zeroth order.

From a Fourier optics perspective, the measured power in spatial errors may be reduced by the frequency response of the interferometer optical system (Goodman 1996; de Groot & de Lega 2006). To quantify this loss and support an analysis of the frequency content of the measured period error, the instrument transfer function (ITF) on the interferometer was independently measured. The ITF is the ratio of the power of a test optic measured by an optical system to the known power present on that test optic as a function of spatial frequency. Hence, the ITF is a measure of how accurately an optical system captures errors at a specific spatial frequency. The ITF can be applied as a corrective factor to provide a more accurate estimate of the spatial frequency content of an optic with unknown spatial frequency content. The ITF of the interferometer system employed was characterized prior to the grating measurements reported here, and is shown in Figure 5.

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The measurement error of the wavefronts is estimated using the repeatability of the interferometer. The grating figure (zeroth 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, as each order also requires a distinct alignment to the interferometer beam. This error is used to assess the error present in our calculation of zeroth order from ± first orders (see Section 3.1) and in the average period error of the grating (see Section 3.2).

As a cross-check on the self-consistency of the height map measurements, we use the height maps of the +first and −first orders to predict the figure of the grating as measured by the zeroth order height map. From Equation (4), we see that:

$$\begin{eqnarray}&&\displaystyle \frac{{{ \mathcal H }}_{+1}(x\cos {\alpha }_{\pm 1},y)+{{ \mathcal H }}_{-1}(x\cos {\alpha }_{\pm 1},y)}{2\cos {\alpha }_{\pm 1}}-{{ \mathcal H }}_{0}(x,y)=0,\end{eqnarray} \tag{ 9 }$$

where zero on the right-hand side of the expression is interpreted in a statistical sense, i.e., is consistent within the error of the interferometer. The left side of Figure 7 displays the result of evaluating the left-hand side of Equation (9) with the measured data. We find that this difference map is centered about zero as expected. Moreover, we find a characteristic variation of 7.0 nm rms across the grating surface, consistent with the variation expected due to interferometer measurement noise (9.8 nm rms). Thus, we conclude that our measurements are consistent with Equation (9), bolstering confidence in the application of the methodology outlined in Section 2.1.

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We next compute the position-dependent groove displacement Δ(x, y) from the height maps of the diffracted orders using Equation (5). The result is shown in Figure 8. The left side of Figure 8 is the displacement map over the whole grating surface, Δ(x, y), while the right side shows several representative traces of groove displacement across the dispersion direction (i.e., Δ(x, y = c), where c is a chosen constant). Note that we have remapped the pixel scale of these ±first order measurements to account for the $\cos {\alpha }_{\pm 1}$ reduction in spatial sampling. In doing so, they can be compared directly to the zeroth order height map (Figure 6), albeit with coarser spatial sampling (0.045 mm pixel⁻¹ versus 0.043 mm pixel⁻¹). Periodic structure in this displacement map on the scale of a couple of millimeters is evident; an analysis of the power spectral density (PSD) of this measurement is performed in Section 3.3.

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From the groove displacement map, the average period error as a function of position is derived using Equation (8). Similar to Figure 8, the left side of Figure 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 the average period error over the entire grating, we construct a histogram of the average period error from the 1.2 × 10⁶ pixels sampling the grating (Figure 10). We assume Gehrels errors in each bin (Gehrels 1986), and fit the resulting distribution with lmfit,⁴ a Python-based curve-fitting package (Newville et al. 2019). We find that the groove period errors are satisfactorily described (${\chi }_{r}^{2}$ = 0.95) by a Gaussian distribution.

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We calculate the rms of the raw average period error map shown in Figure 9(A) to be 0.028 nm. This establishes the correct order of a typical period error over the EBL-written grating; however, it should be both corrected for the impact of the ITF and contextualized relative to the interferometer's measurement error in order to provide a faithful assessment of the groove placement accuracy of EBL.

While the groove period errors are found to be distributed as a uniform Gaussian in magnitude, there is a clear spatial correlation present in Figure 9(A). To elucidate this spatial correlation, we compute the PSD of the period error map in the dispersion direction by averaging the one-dimensional PSDs of each row, i.e., at a fixed groove direction position. As a cross-check on our representative PSD, we calculate the integral of this representative PSD over frequency and find that it agrees with the standard deviation of the groove period error map to within 5% error, as expected via Parseval's theorem.

We next correct this representative PSD with the best-fit ITF from the interferometer calibration (Figure 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 Figure 11. The average period error σ_d as calculated from this PSD is 0.029 nm. Contributing to this average period error is two primary peaks at 0.44 ± 0.02 cycles mm⁻¹ and 4.93 ± 0.02 cycles mm⁻¹, which correspond to spatial periods of 2.27 mm and 0.203 mm, respectively.

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As a point of comparison, we also compute the PSD of the interferometer error, adopting the measurement repeatability as described in Section 2.2 as the error in ${{ \mathcal H }}_{\pm 1}$ and propagating this error through Equations (5) and (8). The resulting PSD from the estimated error is displayed as a red dashed line in Figure 11. Note that we do not correct this error PSD with the ITF, as this is an estimate of the random error inherent in the measurement as opposed to the systematic underprediction of power from the imperfect ITF. We use the error PSD to estimate the ultimate sensitivity of the interferometric measurements described in Section 2.1 by integrating this PSD to calculate a limiting average period error ${\sigma }_{d,\mathrm{noise}}$. Using the interferometer noise floor in this way results in ${\sigma }_{d,\mathrm{noise}}$ of 0.028 nm.

The average groove period error calculated in Section 3 has implications for the adoption of EBL-written gratings by future astronomical spectroscopy missions. Most science applications for grating spectroscopy, such as line detection or assessing velocity fields, have a figure of merit dependent on the spectral resolution R of an instrument, where $R=\lambda /{\rm{\Delta }}\lambda$ and Δλ is the FWHM of the wavelength uncertainty. Following this use of FWHM in the definition of R, we adopt Δd = 2.35σ_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 Gaussian distribution of period errors shown in Figure 10.

The resulting estimate of R for the EBL-written grating tested here is R = d/Δd = 14,600. However, there are two important caveats to this resolution estimate. First, we note that this resolution estimate is dependent on the assumed period d. In general, the period of a grating can be assessed in two ways: (1) the average feature size can be measured with nanometer-scale metrology, such as an AFM or SEM, or (2) under illumination by a pencil beam of known wavelength, the relative angle between diffracted and reflected light can be measured precisely and d calculated by use of Equation (1). However, in both instances, the grating period is assessed locally, i.e., in a small area as compared to the entirety of the grating. Hence, this period may not be representative of the average period over the entirety of the grating, as assumed for the calculation of the average groove period error in Section 3.2. However, recognizing that the overall dependence of R on d is merely d⁻¹, we note that a percent uncertainty in d directly maps to a percent uncertainty 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.

The second, arguably more important, caveat is the issue of the uncertainty inherent in the interferometric measurement. The reported average period error contains contributions from both the true period error of the grating and the interferometer noise floor. As illustrated in Figure 11, the PSD of the interferometric measurement uncertainty is comparable to the PSD of the average period error itself. We therefore argue that the contribution of the interferometer noise 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. For example, the grating's surface quality has not been factored in to arrive at our reported limiting spectral resolution, and requirements on the figure of the grating must be assessed within an instrument context to achieve the desired performance. Nonetheless, our findings indicate that EBL-written gratings are suitable for missions such as Arcus and Lynx given the target spectral resolutions of R > 2500 (Smith et al. 2019) and R > 5000 respectively (Gaskin et al. 2019).

Only two frequency components, 0.44 ± 0.02 cycles mm⁻¹ and 4.93 ± 0.02 cycles mm⁻¹, have significant power above the measured uncertainty PSD. The error quoted for these frequency components is derived from the width of the component in frequency space.

These frequency components occur at spatial scales linked to attributes of the EBL-patterning processes. The high frequency component at 4.93 cycles mm⁻¹ roughly corresponds to the spatial scale of the write field selected for this patterning run, 200 μm. While the measured spatial scale does not correspond directly to the scale of this write field to within 3σ error (4.93 ± 0.02 cycles mm⁻¹ measured versus 5.00 cycles mm⁻¹ expected), the measured frequency is dependent on the interferometer pixel scale calibrated at the time of measurement. A systematic error of 1.2% in this pixel calibration scale is sufficient to explain the difference between the as-measured frequency components and the expected 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 the EBL-writing process. For this particular write, the EBL tool alignment is checked every 7 minutes during the course of the write. During this alignment check, the EBL tool translates to fixed alignment markers patterned on the substrate outside of the grating area and recalibrates its position based on these markers. Examining the tool log, we find that the average separation in the dispersion direction between these calibration points is 2.34 ± 0.09 mm, in agreement with the measured low frequency component.

In both of these alignment processes during the EBL write, the relationship between the sample position and the (fixed) electron column is adjusted. This adjustment introduces stitching errors, or errors in the position of one write field/calibration field to the next. In the context of a grating, these stitching errors would shift grooves contained in one field relative to another. Thus, EBL stitching errors would be expected to result in a discontinuity in groove placement, and appear as a period error occurring at a fixed spatial scale as observed.

To estimate the contribution of each type of stitching error to the average period error, we integrate the measured PSD around the corresponding frequency peaks. The integral from 0.40 to 0.50 cycles mm⁻¹, which encompasses the low frequency peak, contributes σ_{d, f1} = 0.009 nm to the overall power. The high frequency peak contributes σ_{d, f2} = 0.008 nm to the total measured power, as estimated by integrating the PSD over the range of 4.8–5.0 cycles mm⁻¹. Adding these in quadrature, a total of ${\sigma }_{d,\mathrm{EBL}}=\sqrt{{\sigma }_{d,f1}^{2}+{\sigma }_{d,f2}^{2}}=$ 0.012 nm of the measured power is directly attributable to the EBL-writing process itself. We note that no effort has yet been made to minimize the impact of these calibration steps 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 limiting spectral resolution of R ∼35,000. In an astronomical context, this spectral resolution is consistent with the medium-resolution echelle spectroscopy capability of Hubble Space Telescope's Space Telescope Imaging Spectrograph (STIS, Woodgate et al. 1998), albeit with a finer groove period and without the benefit of the STIS cross-disperser.

An important open question is whether the EBL groove placement accuracy as quantified here will translate faithfully to the patterning of more complex grating patterns, e.g., gratings with arbitrary groove orientations or freeform surfaces. Such gratings break the simplifying symmetry of the interferometric measurement technique for constant period, flat gratings as presented in Section 2.1. In principle, however, the interferometric technique can be adapted to assess the expected groove placement accuracy on a customized EBL grating by writing a grating pattern yielding a back-diffracted wave front when illuminated by a plane wave.

To perform such a measurement, a test substrate with a figure representative of the desired customized grating is needed. Assuming a fixed incidence geometry on this test substrate, the local groove density and orientation yielding a back-diffracting wave front can be calculated numerically via a raytracing program. This back-diffracting grating pattern would then be written on a test substrate, and the back-diffracting wave front measured interferometrically. Additionally, detailed knowledge of the figure of the test substrate is required, either from manufacturing metrology or via measurement with a computer-generated hologram. This independent measurement of the test substrate can be used to account for the phase errors introduced in a back-diffracted wave front due to figure. Subtracting the impact of the test substrate figure from the back-diffraction interferogram yields a residual phase error due to imperfect groove placement on this curved substrate.

As a concrete example, we have calculated the customized grating pattern required to assess the groove placement accuracy on a spherical grating. The assumed substrate and incidence geometry for this test grating is shown in Figure 12(A). The sphere is assumed to have a diameter of 76.2 mm and a radius of curvature of 220 mm. The sphere would be 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 wave front assuming this substrate and incidence geometry are shown as the left-hand and right-hand plots in Figure 12(B). Assessing the groove placement accuracy of EBL on a curved substrate such as this simple sphere would serve as a proof-of-concept for the customized gratings discussed in Section 1, and reduce the risk of adoption for instrument concepts featuring these gratings such as the two-element spectrometer (DeRoo et al. 2019).

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