Understanding reliability and some limitations of the images and spectra reconstructed from a multi-monochromatic x-ray imager

Understanding reliability and some limitations of the images and spectra reconstructed from a multi-monochromatic x-ray imager T. Nagayama, , R. C. Mancini, D. Mayes, R. Tommasini, and R. Florido Citation: Rev. Sci. Instrum. 86, 113505 (2015); doi: 10.1063/1.4935828 View online: http://dx.doi.org/10.1063/1.4935828 View Table of Contents: http://aip.scitation.org/toc/rsi/86/11 Published by the American Institute of Physics REVIEW OF SCIENTIFIC INSTRUMENTS 86, 113505 (2015) Understanding reliability and some limitations of the images and spectra reconstructed from a multi-monochromatic x-ray imager T. Nagayama,1,a) R. C. Mancini,1D. Mayes,1R. Tommasini,2and R. Florido3 1Physics Department, University of Nevada, Reno, Nevada 89557, USA 2Lawrence Livermore National Laboratory, Livermore, California 94550, USA 3Departamento de Física, Universidad de Las Palmas de Gran Canaria, 35017 Las Palmas de Gran Canaria, Spain (Received 5 June 2015; accepted 2 November 2015; published online 18 November 2015) Temperature and density asymmetry diagnosis is critical to advance inertial confinement fusion (ICF) science. A multi-monochromatic x-ray imager (MMI) is an attractive diagnostic for this purpose. The MMI records the spectral signature from an ICF implosion core with time resolution, 2-D space resolution, and spectral resolution. While narrow-band images and 2-D space-resolved spectra from the MMI data constrain temperature and density spatial structure of the core, the accuracy of the images and spectra depends not only on the quality of the MMI data but also on the reliability of the post-processing tools. Here, we synthetically quantify the accuracy of images and spectra reconstructed from MMI data. Errors in the reconstructed images are less than a few percent when the space-resolution effect is applied to the modeled images. The errors in the reconstructed 2-D space-resolved spectra are also less than a few percent except those for the peripheral regions. Spectra reconstructed for the peripheral regions have slightly but systematically lower intensities by ∼6% due to the instrumental spatial-resolution effects. However, this does not alter the relative line ratios and widths and thus does not affect the temperature and density diagnostics. We also investigate the impact of the pinhole size variation on the extracted images and spectra. A 10% pinhole size variation could introduce spatial bias to the images and spectra of ∼10%. A correction algorithm is developed, and it successfully reduces the errors to a few percent. It is desirable to perform similar synthetic investigations to fully understand the reliability and limitations of each MMI application. C2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4935828] I. INTRODUCTION Inertial confinement fusion (ICF) is a concept for an alternative energy source, which releases energy by compressing a millimeter-scale capsule containing fusion fuel (i.e., hydrogen isotopes such as tritium, T, and deuterium, D) with ablation pressure driven by mega-joule lasers.1,2 Substantial progress has been made in theory, experiment, and diagnostics of ICF. However, efficient fuel burn, or ignition, has not yet been achieved, and discrepancies between experiments and simulations still remain.3While sources of discrepancies are extensively investigated,4diagnostics that directly reveal the evolution of the ICF implosion-plasma spatial structure are desired to advance ICF science towards ignition. Multi-monochromatic x-ray imagers (MMIs) are attractive instruments for this purpose. The MMI is a unique 2-D spectrometer consisting of a pinhole array, a multilayered mirror (MLM), and a microchannel plate detector (MCP).5,6It records an array of ICF implosion-core images, each of which is formed by photons of slightly different energy. By processing MMI data, one can extract narrowband images (i.e., intensity images of a narrow spectral range), space-integrated spectra,7–9and 2-D space-resolved a)Present address: Sandia National Laboratories, Albuquerque, New Mexico 87185-1106, USA. spectra.10 Spectroscopic analysis of MMI data provides a variety of approaches to constrain ICF plasma spatial structure in electron temperature, Te, and electron density, ne. For example, we mixed a small fraction of Ar into D2 gas fuel and analyzed Ar line emission to characterize Te and neof the core. Using two narrow-band images, we inferred time-resolved 2-D Temaps,11 and from a collection of space-resolved spectra, we extracted neas well as Temaps.10 Synthetic investigations also suggest that time-resolved 3-DTe and nespatial structure of ICF implosion cores can be studied by simultaneously analyzing collections of 2-D space-resolved spectra extracted from three MMI instruments fielded along quasi-orthogonal lines of sight.12 Thus, MMI diagnostics have the potential to better constrain ICF experiments and advance ICF theory. MMI processing consists of two parts: (1) basic processing and (2) post-processing.9,10 The basic processing consists of optical-density to intensity conversion and a series of efficiency corrections associated with mirror reflectivity, filter transmission, flat-fielding, and MCP pore-angle dependent quantum efficiency. The post-processing is the process required to extract narrow-band images and space-resolved spectra after the basic processing has been applied. Because post-processing is numerically more involved, it is often mistaken to be the major source of the analysis inaccuracy. However, this post-processing part is quite reliable as long as it satisfies the reliability criterion discussed in Ref. 13. 0034-6748/2015/86(11)/113505/11/$30.00 86, 113505-1 ©2015 AIP Publishing LLC 113505-2 Nagayama et al. Rev. Sci. Instrum. 86, 113505 (2015) In this article, we quantify the level of inaccuracy introduced into the reconstructed images and spectra due to the post-processing. The case of investigation is its application to the Ar-doped D2ICF implosion experiments. Specifically, we perform synthetic investigations for the spatialand spectralresolution effects. These effects are not obvious because resolution affects the MMI data first and then are propagated to the reconstructed images and spectra through the postprocessing discussed in Refs. 9and 10. In principle, images and spectra could be affected both by spatial resolution and spectral resolution. We found that the reconstructed images are accurate within a few-percent error as long as a point-spread function is applied to the modeled image. The spectral-resolution effects on the image are negligible for our application because the line images are integrated over their spectral line width, which is much broader than the spectral resolution of the instrument. Extracted space-resolved spectra are accurate within a few-percent error for the central region of the image as long as spectral-resolution effects are applied on the modeled spaceresolved spectra. Spectra extracted from the periphery show an average absolute error of ∼6%–7%. The larger discrepancies originate from the systematic intensity drop due to spatial convolution over the image boundary and from periodic artifacts due to the rapid intensity drop at the periphery. Since the intensity drop is systematic and the periodic artifacts oscillate about the expected value, their effects on the line ratio and width are much smaller. The Teand neinferred from those spectra are accurate within a few-percent error. We also investigated individual pinhole size variation and its effect on the images and spectra. This can introduce artificial spatial bias of ∼10%. We developed a technique to correct the pinhole size-variation effects on the MMI data. The correction successfully removes the spatial bias and suppresses the errors to within a few percent. The article is organized as follows. Sec. II describes the spectral model used to create synthetic MMI data and also to compute expected images and spectra. In Sec. III, we quantitatively investigate the spatial-resolution and spectralresolution effects. Sec. IV describes pinhole size-variation effects on the extracted images and spectra, as well as their correction. Sec. Vprovides discussion and summary. II. SPECTRAL MODEL A. Emergent spectra as a function of (x, y) on the image plane Our spectral model computes emergent spectra at every point on the image plane, Iν(x, y), and then Iν(x, y)is used to compute both synthetic MMI data and expected images and spectra. The case of investigation is an OMEGA direct-drive Ar-doped D2ICF implosion.10,11 First, local emissivity and opacity of Ar are computed for ranges of Teand ne, and then, the emergent spectra, Iν(x, y), are computed by solving the radiation transport along each chord, z, parallel to the line of sight.12 To compute the Ar emissivity, ϵν, and opacity, κν, necessary atomic data are computed by Flexible Atomic Code (FAC),14 and Ar level populations are solved for grids of Teand newith a collisional-radiative model, ABAKO.15 We used the Stewart and Pyatt model16 for continuum-lowering effects and the escape factor of a sphere to account for photopumping effects on the populations.17 Spectral emissivity and opacity are computed from the calculated populations taking into account detailed Stark line profiles computed by MERL.18 Emergent spectra, Iν(x, y)[erg/s cm−2sr−1eV−1], are computed by numerically integrating the following radiation transport equation along chords in the source assuming parallel ray tracing,19 dIν(x, y, z) dz =ϵν(x, y, z)−Iν(x, y, z)κν(x, y, z).(1) If (x, y, z)is inside the object, the quantities ϵν(x, y, z) and κν(x, y, z)are determined at the given Teand neusing ABAKO. If (x, y, z)is outside the object, both ϵν(x, y, z) and κν(x, y, z)are zero. Theoretical uncertainty in the calculated emissivity and opacity does not affect the conclusions derived from our synthetic investigations because Iν(x, y)is used to calculate both synthetic MMI data and expected images and spectra. What we pursue is an understanding of the discrepancies due to the processing of MMI data, whose spatialand spectralsamplings are somewhat limited. B. Expected images and spectra Expected narrow-band images are computed by integrating Iν(x, y)over any given spectral range as I∆ν(x, y)=∆ν Iν(x, y)dν, (2) where ∆νis the bandwidth of interest. Space-resolved spectra are computed by integrating Iν(x, y)over any given spatial region of interest Iν,∆A=∆A Iν(x, y)da,(3) where ∆Ais an arbitrarily shaped area of interest. There are two different ways to deal with the instrumentalresolution effects. One is to deconvolve the resolution effects from the data. The other is to convolve resolution effects into the modeled images and spectra. We use the latter option because it is more straightforward and robust. For example, we can apply a 2-D Gaussian point-spread function to the expected images to account for instrumental spatial-resolution effects. Also, a 1-D Gaussian spectral profile can be applied to the expected spectra that account for instrumental spectralresolution effects on the spectra. C. Synthetic MMI data Fig. 1(a) shows example MMI data. The detector plane is packed with a hexagonal array of pinhole implosion-core images. Due to the reflection of the MLM, each column of the MMI data picks out signals of specific photon energy, and the horizontal axis is also the spectral resolution axis. 113505-3 Nagayama et al. Rev. Sci. Instrum. 86, 113505 (2015) FIG. 1. (a) MMI data after efficiency correction. iand jare the horizontal and vertical indices of the pixels, respectively. The horizontal axis is also photon energy due to the spectral dispersion by the MLM. Red dots mark the pinhole-image centers. (b) Blow-up of one pinhole image. Each image has its own local spatial coordinates of the implosion core, (x, y). Each pixel value is proportional to Iν(x, y)∆hν(∆L)2. Fig. 1(b) is a blow-up of one pinhole image. After correcting for magnification, each pinhole image has its own local spatial coordinate of the implosion core, (x, y). Thus, each pixel of the MMI data, MMI[i,j], represents signal at a local position of the implosion core surface, (x± ∆L/2, y±∆L/2), of a very small spectral range, hν±∆hν/2. To model MMI data, the value of each pixel, MMI[i,j], is computed as Iν(x, y)∆hν(∆L)2, where ∆hνand ∆Lare the spectral and spatial width of the pixel, [i,j](i.e., not the resolution of the instrument). The mapping from the MMI pixel [i,j]to the corresponding point in the object space, (x, y), and the corresponding photon energy, hν, are different for every dataset and determined through the processing described in Ref. 9. Synthetic investigation becomes more useful when the mapping information comes from the actual data of interest. Then, the spatialand spectral-resolution effects are applied on the synthetic MMI data. Spatial resolution due to the pinholes is applied by convolving a 2-D Gaussian point-spread function of a spatial resolution given by pinhole size. Spectral-resolution effects are applied by convolving the synthetic MMI data with a Gaussian profile of a given resolution power along the horizontal direction. The model also has an option to simulate pinhole sizevariation effects due to an individual pinhole tolerance. When each pixel has the same pinhole size, the following 2D Gaussian convolution is used to account for the spatial resolution effect: M MIconv[i,j]=A−1 i′j′ [M MI [i′,j′] ×exp   −(i′−i)2+(j′−j)2 2σ2 p          ,(4) A= i′j′ exp   −(i′−i)2+(j′−j)2 2σ2 p   , σp=Dp M+1 M 1 2√2 ln 2 1 ∆L, where A−1is the normalization constant and σpis the Gaussian convolution width. Dpis the pinhole diameter, a factor (M+1)/Mconverts Dpto the full-width of the convolving area on the object, a factor 1/2√2 ln 2 converts the full-width to the Gaussian width, and 1/∆Lconverts the units of the width to pixels. In reality, each pinhole has slightly different size than the nominal size DPH due to the fabrication tolerance ∆DPH. This produces variations in both brightness and convolving width of each pinhole image of the MMI data, which are numerically treated as follows. 1. Assign different pinhole size, Dp, to each pinhole, p, using a Gaussian random number generator based on a given nominal pinhole diameter, DPH, and its tolerance, ∆DPH. 2. For a given pixel [i,j]of the synthetic MMI data, (a) find the pinhole pto which this pixel belongs and find its assigned diameter, Dp; (b) compute the convolution with a modified 2-D Gaussian using the assigned Dp, M MIconv[i,j]=B(Dp)A−1 i′j′ [M MI [i′,j′] ×exp   −(i′−i)2+(j′−j)2 2σ2 p          , (5) where Dpis the diameter of the pinhole to which this pixel belongs, σpis the Gaussian width for this particular pinhole, and BDp=D2 p/D2 PH is the simulated relative brightness with respect to the nominal. This simulates the effects of pinhole size variation in brightness and in spatial resolution. The pinhole shape is assumed to be circular, and the effect of the pinhole shape variation is neglected throughout this article. III. QUANTITATIVE INVESTIGATIONS OF SPATIAL AND SPECTRAL RESOLUTION EFFECTS The accuracy of the narrow-band images and spectra reconstructed from MMI data are quantitatively investigated using synthetic MMI data. We create synthetic MMI data that satisfy the criterion discussed in Ref. 13. We assume a 100-µm-diameter spherical plasma of uniform conditions 113505-4 Nagayama et al. Rev. Sci. Instrum. 86, 113505 (2015) (Te=1500 eV and ne=1.5×1024 cm−3). These values are characteristic values for the OMEGA direct-drive Ar-doped ICF implosion experiments reported previously.10,11 The pinhole-array design and photon-energy axis used for the synthetic MMI data are those from the data shown in Ref. 11 that satisfy the criterion. Then, we apply two different levels of instrumental details by following the technique discussed in Sec. II C. The first case, denoted as “∆E,” takes into account spectral resolving power of E/∆E=150. The second case denoted as “∆E,∆x,” takes into account both the spectral resolution and the spatial resolution (∆x=11 µm). These are typical values for the MLM and pinhole arrays used in our applications. The images and spectra are reconstructed from these synthetic MMI data9,10 and compared with the expected images and spectra computed as in Sec. II B. Comparisons of the images and spectra are discussed in Sec. III A and in Sec. III B, respectively. A. Accuracy of reconstructed images Fig. 2compares Ar He-βimages reconstructed from synthetic MMI data for cases (a) “∆E” and (b) “∆E,∆x” with (c) the expected He-βimage. The reconstructed images for both cases agree with the expected image quite well. However, they show small artificial structures along the ydirection, which correspond to the vertical direction in the MMI image and originate from the discrete nature of MMI data. These artifacts of a few percent in value are unavoidable and are limitations of the current MMI data and their processing technique. The reconstructed image from the MMI data without spatial-resolution effects [i.e., Fig. 2(a)] best agrees with the expected images with an average absolute error of 1.5%. When the spatial-resolution effects are applied to the synthetic MMI data, the average absolute error increases to 7.1%. Figs. 3(a) and 3(b) show percent-error surface plots computed for “∆E” and “∆E,∆x” with respect to the expected image, 2(c). While Fig. 3(a) reveals that the main source of error is from the observed vertical artifacts, the errors in (b) are largest at the image periphery. These large errors at the periphery are produced because the expected image shown 2(c) does not take into account spatial-resolution effects. Next, we apply the 2-D Gaussian point-spread function of FWHM =11 µm directly to the expected image and re-compute the error image for the case of “∆E,∆x” to see if such an image is a more appropriate representation of the reconstructed images. We note that this treatment does not necessarily compensate for the spatial-resolution effects on the reconstructed images. This is because they are produced by applying the point-spread function and spectralresolution effects directly to the MMI data, and then the effects are propagated through the processing. Fig. 3(c) shows the resultant percent-error image. The deep negative errors at the periphery disappear, and the percent error is dominated by the vertical artifacts as in the case of Fig. 3(a). The percent error is reduced from 7.1% to 1.4%. To summarize, the image reconstruction is reliable with a few-percent error as long as the spatial-resolution effects FIG. 2. Ar He-βimages reconstructed from synthetic MMI data with (a) spectral resolution (E/∆E=150), and (b) both spectral and spatial resolutions (∆x=11 µm). (c) Expected image computed directly from the spectral model. are taken into account on the expected image. On one hand, the spectral-resolution effects on the reconstructed image are negligible for the applications presented here because the narrow-band images are computed by integrating a 113505-5 Nagayama et al. Rev. Sci. Instrum. 86, 113505 (2015) FIG. 3. Percent-error surface plots for He-βreconstructed from synthetic MMI data (a) with spectral resolution only and (b) with both spectral and spatial resolutions. The given percentages are averages of the absolute percent errors within the image regions. (c) is the same as (b) except that spatial-resolution effects are directly applied to the expected image used in the percent-error calculation. monochromatic image over its spectral line width (i.e., ∆E∼ 60 eV) as in Eq. (2), which is significantly broader than the spectral resolution of the instruments. On the other hand, the spatial-resolution effects are not negligible and mostly affect the image periphery. The intensities at the image periphery are lowered because the convolving area partially exceeds the object boundary. Small vertical errors observed in Figs. 3(a) and 3(c) originate from the subtle vertical artifacts observed in Fig. 3, which are produced by the discrete nature of the MMI data. B. Accuracy of reconstructed spectra Space-resolved spectra are reconstructed from the synthetic MMI data with the technique discussed in Ref. 10. Spatial regions defined for the space-resolved spectra are shown in Fig. 4(a). Each spatial region can be categorized as belonging to the (b) central region, (c) top/bottom periphery, and (d) left/right periphery as shown in Fig. 4. These categories are important for understanding different sources of errors in MMI spectra. From the particular synthetic MMI data, 53 spatial regions are automatically defined using a minimum binning width comparable to the spatial resolution, ∼11 µm. In this section, one spectrum is selected from each category (i.e., regions 30, 3, and 26, respectively) to discuss the source of error, and the rest are discussed through the average percent errors for central and peripheral regions. Fig. 5(a) compares the reconstructed space-resolved spectra and the expected spectra for a central region [i.e., region 30 of Fig. 4(a)]. The solid red and dashed green lines are those reconstructed from synthetic MMI computed with “∆E” and “∆E,∆x” resolution options, respectively. They agree very well with the expected spectrum (i.e., dotted black). The average percent errors for “∆E” and “∆E,∆x” are 0.8% and 0.2%, respectively. Fig. 5(b) shows similar comparisons for a top/bottomperiphery region (i.e., region 3 of Fig. 4). While they still show reasonable agreement to the expected spectrum, the average percent errors increase to 2.6% and 5.5%, respectively. The increase in the percent error is mostly due to the non-negligible intensity gradient within the region. Spatial areas are defined such that the region size becomes comparable FIG. 4. (a) Rectangular spatial regions defined with spatial resolution, ∆x∼ 11 µm. (b)–(d) define the central region, top/bottom periphery, and left/right periphery, respectively, which have slightly different sources of discrepancies. 113505-6 Nagayama et al. Rev. Sci. Instrum. 86, 113505 (2015) FIG. 5. 2-D space-resolved spectra extracted from synthetic MMI with (solid red) spectral-resolution effects only, and (dashed green) both spectral and spatial resolutions for (a) the central region, (b) the bottom periphery, and (c) the left periphery. The percentages shown are the average absolute percent errors with respect to the expected space-resolved spectra (dotted black). to the spatial resolution so that the intensity variation over the region becomes negligible. However, image intensity always drops rapidly towards the image periphery, and negligible variations cannot be assumed for the peripheral regions. Thus, comparison between expected and reconstructed spectra from the peripheral regions is in general not as good as those for central regions. Spectra extracted from the periphery have other issues when finite spatial resolution is taken into account (i.e., dashed green line). Finite spatial resolution smooths out the structure, and some signals are mixed in from the adjacent regions. Thus, at the periphery, spatial-resolution effects lower the overall intensity very slightly, but systematically, because the convolving area partially exceeds the object boundary. Since the spectral-resolution size is ∼11 µm and small compared to the object size, these effects are subtle but still persist as observed in Fig. 3(b). In the case of image comparison, the systematic intensity drop at the periphery disappears by applying the point-spread function directly to the expected images. However, it is not as easy to take into account spatialresolution effects in the space-resolved spectra modeling. Here, we do not introduce any correction associated with the edge effects but just summarize what we learn from the comparisons. Fortunately, their effects are small and systematic over the entire spectral range. This preserves the line ratios and widths, and thus these effects on the Teand ne analysis can be considered negligible. Fig. 5(c) shows similar comparisons for a left/rightperiphery region (i.e., region 26). One can observe that all the extracted space-resolved spectra show periodic structure about the expected spectrum, which results in increasing the percent errors to 8.2% and 6.5%, respectively. This periodic structure is observed in all space-resolved spectra extracted from the left/right-periphery regions. These periodic structures originate from the non-negligible horizontal intensity gradient in the spatial region. For left/right-periphery regions, the intrinsic radial intensity gradients become horizontal, which is a problem because the horizontal axis is also the spectral axis. Figs. 6(a) and 6(b) show a blow-up of the partial MMI data associated with region 26 over the Lyβregion and the extracted spectrum, respectively. Each spatial region of Fig. 6(a) is slightly enlarged, and outside of region 26 is whitened out for communication purposes. Every region 26 of the synthetic MMI data shows horizontal intensity gradients monotonically increasing from left to right. Since each partial image is responsible for a different sub-range of the spectrum, the extracted spectrum inherits the periodic horizontal gradient structure. While we apply a first-order correction to this using the technique discussed in Sec. V of Ref. 9, the periodic structure is not perfectly removed. Table Ishows average percent errors computed for spaceresolved spectra over each category defined in Figs. 4(b)–4(d). For the 32 central regions, extracted spectra are very accurate and show average percent error of less than 1%. Spectra extracted from the top/bottom periphery show larger discrepancies of 1.4% and 3.8%, respectively. The 1.4% is due to the non-negligible vertical intensity gradient. When the spatialresolution effect is introduced, it slightly but systematically lowers the intensity, which results in increasing the percent error to 3.8%. For the left/right periphery, the percent errors become even larger—5.1% and 7.5%, respectively—due to 113505-7 Nagayama et al. Rev. Sci. Instrum. 86, 113505 (2015) FIG. 6. (a) A left-periphery region (region 26) is picked out from each pinhole image of the MMI data and shown over the spectral range of Ly-β. (b) The spectra extracted from the partial MMI data show periodic structure originating from the repeating horizontal intensity gradient structures of the each rectangular region. The selected regions are slightly enlarged for display purposes. the periodic structure originating from horizontal intensity gradients. Table II shows the percent errors in inferred Teand ne and their standard deviations. Since we have both expected spectra and reconstructed spectra, we can investigate how the inaccuracies in the reconstructed spectra affect the inferred Te TABLE I. The percent errors averaged over the central, top/bottomperiphery, and left/right-periphery regions, respectively. The numbers of spectra are shown in parentheses. For peripheral regions, the percent error becomes larger when spatial-resolution effects are introduced. Left/right peripheral regions have larger errors than top/bottom peripheral region due to periodic structure introduced by the horizontal intensity gradients. Intensity % error ∆E(%) ∆E,∆x(%) Central (32) 0.6 ±0.3 0.7 ±0.4 Top/bottom periphery (5) 1.4 ±0.3 3.8 ±0.3 Left/right periphery (16) 5.1 ±1.2 7.5 ±1.9 TABLE II. The percent errors in Teand neaveraged over the central, top/bottom-periphery, and left/right-periphery regions, respectively. The errors are within a few percent. Te% error ∆E(%) ∆E,∆x(%) Central (32) 0.0 ±0.0 0.1 ±0.1 Top/bottom periphery (5) 0.4 ±0.5 0.5 ±0.6 Left/right periphery (16) 0.7 ±0.6 0.8 ±0.7 ne% error ∆E(%) ∆E,∆x(%) Central (32) 0.2 ±0.2 0.3 ±0.3 Top/bottom periphery (5) 2.3 ±1.1 3.1 ±1.2 Left/right periphery (16) 1.8 ±2.6 1.9 ±2.1 and ne. First, Teand neare inferred from both expected spectra and reconstructed in the same way as described in Ref. 10. The percent errors in inferred Teand neare computed for all 53 regions and averaged over central, top/bottom peripheral, and left/right peripheral regions. In spite of the noticeable discrepancies in reconstructed spectra shown in Fig. 5, they do not significantly affect the line ratios and widths and their impacts on the inferred conditions are all within a few percent. FIG. 7. Percent-error surface plots for reconstructed He-βimages (a) before and (b) after applying the pinhole size-variation correction to the synthetic MMI data. Spatial-resolution effects are taken into account in the expected images. 113505-8 Nagayama et al. Rev. Sci. Instrum. 86, 113505 (2015) To summarize, the dominant source of discrepancies for the peripheral regions is (i) systematic intensity lowering at the periphery due to spatial-resolution effects and (ii) the periodic structure due to horizontal intensity gradients. IV. PINHOLE SIZE-VARIATION EFFECTS AND THEIR CORRECTION Each pinhole of the pinhole array has a slightly different shape and size due to fabrication tolerance. As discussed in Sec. II C, pinhole size-variation introduces variation in brightness and in spatial resolution from one pinhole image to another. While the pinhole shape also affects the MMI data, its effect is neglected throughout this article. In this section, we simulate the impact of the pinhole size-variation effects on the reconstructed images and spectra. We then introduce a correction and discuss how well this correction removes the introduced bias from the reconstructed images and spectra. The uncertainty of the pinhole location is less than a few microns. This is much smaller than the spatial resolution of the instrument and thus not considered throughout this investigation. First, we create synthetic MMI data with pinhole sizevariation effects by following the technique discussed in Sec. II C. The pinhole size and its tolerance used in this synthetic study are 10 ±1µm, which are typical values for the pinhole array in our application. Both spectral resolution and spatial resolution are taken into account in the synthetic MMI data. Fig. 7(a) shows the resultant percent-error surface plots for reconstructed Ar He-βimages. The spatial-resolution effect is applied to the expected image used in the percenterror calculation as discussed in Sec. III A. Compared to Fig. 3(c), the overall error increases from 1.4% to 5.0%. More importantly, it introduces a bias in the spatial shape. There is a larger error band (∼10%) along the ydirection. We confirm that the signal of this bright region comes from a single pinhole image in the synthetic MMI data whose pinhole size is about 5% larger than the nominal and whose intensity is B(Dp)=(1.05)2≈10% brighter than the nominal. The solid green lines in Fig. 8are space-resolved spectra reconstructed from synthetic MMI data with pinhole size variations for (a) a central region, (b) a bottom peripheral region, and (c) a left peripheral region. Compared to the reconstructed spectra without pinhole size-variation effects in Fig. 5, the overall percent errors increase from (a) 0.2% to 8.8%, (b) 5.5% to 8.7%, and (c) 6.5% to 9.3%, respectively. For peripheral regions, one might think that the impact is not as large since the errors without pinhole size-variation effects are already showing ∼6%. However, percent errors without pinhole sizevariation effects are systematic without changing the line ratios and widths [i.e., the green lines in Fig. 5], while percent errors due to the pinhole size-variation effects could affect the line ratios and the widths. Thus, from a diagnostics point of view, the errors due to the pinhole size-variation effects are more important than those due to the systematic intensity lowering and periodic structure discussed in Sec. III B. The introduced bias observed in this synthetic investigation is a concern for the actual MMI data analysis. For example, if one of the pinholes appearing at a line center happens to have a size larger than the vendor’s tolerance, this could introduce significant spatial bias to the extracted images FIG. 8. Space-resolved spectra over the He-βand Ly-βspectral range (solid green) before and (dashed red) after the pinhole size-variation correction for (a) a central region, (b) a bottom peripheral region, and (c) a left peripheral region. The dotted black spectra are the expected spectra.