Hitomi,1,2 the sixth Japanese mission formerly known as “ASTRO-H,”3 was equipped with x-ray focusing instruments to perform imaging and spectroscopic studies in a wide energy range from 0.3 to 80 keV. The x-ray optics on board Hitomi consists of two Hard X-ray Telescopes (HXT)4,5 and two Soft X-ray Telescopes (SXT).6,7 These four X-ray Telescopes (XRTs) employed conical approximation of the Wolter-I type optics and a nested thin-foil structure to enhance their effective areas. In combination with the SXT and x-ray microcalorimeter,8 Hitomi allowed us to perform x-ray spectroscopy in the 0.3- to 12-keV band with unprecedented high-energy resolution of at 6 keV,9 while the other SXT and x-ray CCD camera10 provided a wide field-of-view image () in the 0.4- to 12-keV band. The HXTs, together with the hard x-ray imagers (HXI),11 having a field of view of , enabled us to carry out hard x-ray (5 to 80 keV) imaging spectroscopy. The top-level requirements for the HXTs to achieve scientific goals of the Hitomi satellite are summarized in Table 1.
Top-level requirements for the HXTs.
The HXT is composed of a mirror module, a precollimator,12 two thermal shields (TSs),13 and an alignment cube mirror. A couple of pictures of the HXT module are shown in Figs. 11 and 14 of Ref. 5, and the cross-sectional drawing is shown in Fig. 2 of the same paper. The mirror module is divided into three identical segments. In each segment, 213 primary–secondary pairs of Pt/C multilayer-coated thin (0.22 mm) foils are inserted. We assign numbers as #1, #2, …, #213 from the innermost foil to the outermost one. The top and bottom edges of the foils are supported with alignment bars. Each segment has nine sections that are partitioned with these alignment bars. We designate an aperture between two neighboring alignment bars as a “sector.” The precollimator is installed on the front end of the mirror module to reduce stray light, while the TSs are attached on top and bottom of the mirror + precollimator modules. Each TS consists of an aluminized PET film with thickness supported with a stainless steel mesh and an aluminum frame (see Fig. 1 of Ref. 13). These top/bottom TSs mitigate radiation coupling and help keep the HXT temperature in a range of in orbit. The alignment cube mirror is used to record the x-ray optical axis of the mirror. The detailed design of the HXT is described in Ref. 5, in which the authors also mention the tuning process of the x-ray images and the x-ray characteristics above 20 keV of the HXTs, both of which were achieved at SPring-8, one of the Japanese synchrotron radiation facilities. The measurements of the on-ground HXT calibration at SPring-8 were performed in November 2012 for the HXT-1 and December 2013 for the HXT-2. To evaluate the x-ray performance below 20 keV, we also carried out the soft x-ray measurement using the beamline facility at Institute of Space and Astronautical Science (ISAS)/Japan Aerospace eXploration Agency (JAXA) in September 2014.
The data obtained during the on-ground calibration are a baseline of the HXT performance that was used to determine parameters stored in calibration files for a ray-tracing simulator in the Hitomi response generator (Yaqoob et al.,14 in preparation). The IGR J16318–4848, G21.5–0.9, and Crab data, taken during the initial operation phase, were used to evaluate the in-orbit HXT performance and to compare the outputs of the response generator. The in-orbit calibration and its results are summarized in Matsumoto et al.15 Therefore, in this paper, we focus on the comprehensive result of the on-ground calibration, except for the x-ray characteristics of the backside reflection, e.g., the x-ray reflectivity of bare aluminum substrates in the high-energy band. Since the on-ground measurements allow us to examine the local performance of the HXTs, we present here the analysis to figure out key ingredients that govern the angular resolution and effective area of the HXTs.
This paper is organized as follows: first, we mention briefly the measurement system in the BL20B2 beamline at SPring-8 and the ISAS/JAXA 27 m beamline in Sec. 2. In Sec. 2, we also describe methods of the data acquisition used in the respective beamlines. The results of the HXT on-ground calibration including the angular resolution, effective area, vignetting functions, and stray light are described in Sec. 3. The investigation of the key factors that cause the degradation of the angular resolution and the effective area is explained in Sec. 4. A summary of this paper is given in Sec. 5.
Experimental Facilities and Data Acquisition
SPring-8 BL20B2 Beamline
As a frequent user group, we developed the measurement system that can be utilized generally for hard x-ray mirrors in the BL20B2 beamline.16 The BL20B2 beamline covers a wide energy range of 5 to 113 keV and provides monochromatic x-rays produced by a double-crystal monochromator of for x-ray imaging studies, suitable for the calibration on the performance of the HXT. As a power user application (PI: Kunieda, 2009 to 2013), we utilized the BL20B2 beamline for about 3 weeks per mirror to carry out the on-ground calibration. This beamline was also utilized to verify and validate the x-ray reflectivity of several foils and to establish a method of mirror tuning.5,17 The detail of the measurement system and data acquisition, including the specification of the BL20B2 beamline, is described in Ref. 16. Thus, only a brief overview is given in the following three paragraphs.
Figure 1 shows a schematic view of the measurement configuration, including the definition of the coordinates. Both the HXT and the detector were mounted on the respective movable translation stages. These stages moved in the horizontal () and vertical () directions perpendicular to incident x-rays. In addition, the HXT stage could rotate around the axes. A movable four-jaw slit was placed between a transport duct and the HXT stage; it allowed a change in the beam size in the - and -directions. The x-ray beam collimated to was commonly used in the measurement to ensure parallelism of , which is determined by the beam size and the distance between the x-ray source and the slit (), independent of the actual spot size of the source. This parallelism is high enough to evaluate the angular resolution of the HXTs. The distance between the HXT and a detector was set to 12.761 m, taking into account the beam divergence, which is slightly longer than the nominal focal length (12 m). We note that the temperature in the experimental hutch was controlled to be during the measurement.
As a wide field-of-view imaging detector, Image Intensifier C7336 (hereafter I.I.) + optical CCD camera C4880 (Hamamatsu Photonics K.K.) with a focusing lens was used. The CCD camera has , and the plate scale is in combination with I.I. Thus, the detector has a field of view of , corresponding to on the focal plane. When using the I.I., we inserted one 1% ND filter between the I.I. and the CCD camera to avoid saturation. For the mirror alignment or the active tuning of the alignment bars supporting the mirror foils, we also used a scintillator, instead of the Image Intensifier C7336. Since the plate scale was changed to be , the scintillator + CCD camera allowed us to carry out the fine tuning.
To illuminate the full aperture of the HXT by the beam, we moved the HXT and detector stages synchronously. While the HXT stage was moved with a 10-mm pitch in the -directions, the detector stage was done with a 10.63-mm pitch; the amount of the movement was corrected with so that the focused spot of the reflected x-rays could be located at the same position, i.e., the center of the detector. This method is called a pointing scan. Due to the limitation of the movable range of the translation stages, only one segment could be fully measured at one pointing scan (see Fig. 16 of Ref. 5). We can obtain a line-like x-ray snapshot at each pointing; examples of the snapshots are shown in Fig. 2. A surface brightness profile of each snapshot depends on the local performance within the x-ray illuminated area. We note that each snapshot was taken 2 to 6 times to increase photon statistics. After taking the snapshot, a mechanical shutter placed on the beam path was closed to take a corresponding dark frame. A total of 522 snapshots were obtained per segment. For the measurements of the other two segments, the mirror was rotated around the -axis by . These x-ray snapshots obtained at the respective pointings were summed together to reconstruct a focal plane image; the resulting image is analyzed to evaluate the effective area and angular resolution of the whole mirror.
Before measuring the performance of the HXT, that of the detector system was examined as well, especially the I.I. + optical CCD camera. Since the sensitivity of the I.I. is not uniform, we made a flat-field image as described in Appendix A. The flat-field correction was applied to each snapshot. In addition, since optical light created in the last stage of the I.I. is reflected multiple times between the focusing lens and the CCD chip, ghost images are produced in the focal plane image, which then contaminates the focused x-ray image [see Fig. 2(d)]. Although it is difficult to identify the causes of the ghost images, we found that a dim ellipse, called a “halo,” always appeared around the x-ray images. Thus, we made narrow masks to extract the reflected x-rays from the focal plane images. The mask was applied to the corresponding x-ray snapshot to remove a significant part of the halo.
ISAS/JAXA 27-m Beamline
For energies below 20 keV, x-rays get severely absorbed by air. Thus, to evaluate the HXT performance in the soft x-ray band (), we used the 27-m beamline facility at ISAS/JAXA. The ISAS/JAXA beamline is equipped with a vacuum chamber that can accommodate x-ray mirrors and focal plane detectors altogether; the physical limits of the size of the diameter and the focal length of the mirror are 450 mm and 9 m, respectively. The on-ground calibrations of the XRTs on board ASCA,18 Astro-E1,19 Suzaku,20 and ASTRO-H21,22 have been carried out so far in the ISAS/JAXA beamline. The beamline was upgraded recently for the on-ground calibration of the ASTRO-H SXTs. The detail of the current beamline system including the detectors is described in Ref. 23.
We used a proportional counter with an aluminized-mylar window of 12-mm diameter and a back-illuminated x-ray CCD camera made by Hamamatsu Photonics K.K. as focal plane detectors. The CCD camera has with its pixel size of . These detectors were mounted on the translation stages. We adjusted the position of the stage so that the distance between the HXT and these detectors was set to 6 m, a half of the HXT focal length. We illuminated the HXT aperture by a beam; the beam divergence was 23″. The HXT was moved for the x-ray beam to sweep the HXT aperture in the -direction. The detector stage was also moved synchronously with a half speed of the HXT stage to realize the same configuration of the HXT and the detector as that at SPring-8. After a one-line sweep, the HXT and detector stages were moved in the -direction with a pitch of 4 mm, and then the next sweep was conducted. This measurement method is called a raster scan. We note that the x-ray beam covered of the HXT aperture in this raster scan.
Similar to the pointing scan, we reconstructed the focal plane image by adding together a snapshot of each sweep taken with the x-ray CCD camera. Since the detector position was placed at 6 m from the mirror, the reflected x-rays would reach the detector before converging; the beam would converge effectively to on the detector. As a result, the core of the focal plane image would be blurred. We also note that the effective spot size at the x-ray generator is . Therefore, we focused only on the evaluation of the effective area, half-power diameter (HPD), and stray light in this system. We used characteristic x-rays of 4.51 keV (), 8.04 keV (), and 17.8 keV () in the measurement. While the and x-rays were monochromatized by a double-crystal monochromator, we used a Ti filter with a thickness for the measurement.
During the measurement, we monitored the temperature of the chamber with -type thermocouples. In addition, to avoid contaminants caused by malfunction of the vacuum system, we attached two contamination covers consisting of polypropylene film with a thickness of on both the top and bottom sides of the HXTs. The contamination was also monitored with thermoelectric quartz crystal microbalance (TQCM). We found that the changes of the oscillation frequency of the TQCM were below 5 Hz during the measurements, corresponding to adsorption of the contaminants.
Results of the On-Ground Calibration
Vignetting Functions and Optical Axes
We first determined the optical axis, at which the maximum throughput is obtained, by measuring the angular dependence of the effective area. The measurement was conducted by tilting the mirror stage around the - and -axes, respectively, and then defining the angles in and , where the optimal value in the effective area was attained. We designate this direction as “on-axis” hereafter. If the mirror is tilted against the optical axis (hereafter, “off-axis”), the x-ray path for the normal double reflection of each foil pair is intercepted by the neighboring foils. Furthermore, the x-ray incident angles for some parts of the foils become larger than the critical angle of reflection; hence, it leads to reduction in the x-ray reflectivity. These effects, so-called vignetting, cause a decrease in effective areas at off-axis angles. In the BL20B2 beamline, the off-axis dependence of the effective areas (hereafter, “vignetting curve”) was measured at 50 keV. Since the HXTs are sensitive to the vignetting effect at higher x-ray energies, the vignetting curve would be more sharply peaked at 50 keV, and therefore the optical axis can be determined more accurately. For the HXT-1, we also measured the vignetting curve at 30 keV. In this measurement, the translation stage for the mirror was moved with a 20-mm pitch to save time; the beam-illuminated area is only 25% of the whole mirror aperture. We note that the on-axis effective area obtained with of the mirror illumination is consistent with that with the full illumination within accuracy.
As an origin of , we defined a mechanical axis using fiducial structures of the mirror housing and then measured the effective areas of each segment at off-axis angles from the mechanical axis of and . Because of the measurement configuration, each segment is more sensitive to the vignetting effect around the -axis than that around the -axis. Thus, the pitch angles were set to 0.5′ for and 1′ for . We modeled the resultant vignetting curves with Gaussian + Lorentzian functions as follows:24 this model is an empirical formalization to reproduce well the core and tail of the vignetting curves. The smooth interpolation by the model fit to the limited data points provides a good estimation of effective areas at arbitrary off-axis angles.
For the HXT-1, the vignetting curves of segment 1 are shown in Fig. 20 of Ref. 5. Using the best-fit vignetting curves of the ’th segment around the - and -axes, and , we formulated the effective area of each segment () as a function ofFig. 3. The color map is normalized by the on-axis effective area of each mirror. The origin of the color map represents the mechanical axis. The optical axes are found at for the HXT-1 and for the HXT-2.
Moreover, we measured the vignetting curves at 8 keV in the ISAS beamline. The vignetting curves along the - and -axes are shown in Fig. 4. We note that the data points at 8 keV were well reproduced by Gaussian + Lorentzian functions with an accuracy of . Using the color maps in Fig. 3, we calculated the mirror vignetting at 30 or 50 keV; for example, we extracted the profiles at for the HXT-1 and for the HXT-2 to obtain the vignetting curve in the -direction. The resultant vignetting curves are also shown in Fig. 4. We note that the optical axes at 8, 30, and 50 keV were all consistent within an accuracy of 15″ between the SPring-8 and ISAS/JAXA measurements.
The fields of view, defined by a full-width half maximum of the vignetting curve, are summarized in Table 2. The fields of view at 8, 30, and 50 keV were estimated from the model curves in Fig. 4 to be 9.7′ (both HXT-1 and HXT-2), 7.7′ (HXT-1), and 6.4′ (HXT-1)/6.1′ (HXT-2), respectively, slightly larger than those estimated from a ray-tracing simulation for the ideal case (see Fig. 6 of Ref. 5): 6.4′ (30 keV) and 5.6′ (50 keV). (The fields of view of at 30 keV and at 50 keV are estimated from the latest ray-tracing simulator used in the Hitomi software. The comparisons of the vignetting curves between the data and the simulations are shown in Figs. 13 and 14 in the Hitomi calibration document, Ref. 25. The positioning errors and figure errors of the foils, which are discussed later, are implemented in the calibration file for the simulator. The consistency of the fields of view also supports that the key factors to the HXT performance are both positioning and figure errors.)
Summary of the HXT fields of view.
On-Axis Angular Resolution and Effective Area
Figures 5 and 6 show on-axis focal plane images of the HXTs taken at the energies of 20, 30, 40, 50, 60, and 70 keV, respectively. We also show the x-ray images at 8 and 17.8 keV in Fig. 7. The 8 and 17.8 keV images were taken with the detector placed only 6 m away from the mirror. Since the focal position is extended to 21.6 m in the ISAS/JAXA 27 m beamline, this detector position means that we took off-plane images at 8 and 17.8 keV. Thus, the core of the image was severely blurred. However, we can still use the defocused images to evaluate the effective area and the angular resolution except for the image core. (The integrated region for the effective area is much larger than the scale of the image blurring.)
The point spread function (PSF), which represents the surface brightness profile when observing an x-ray point source, is shown in Fig. 8. Here, we averaged the surface brightness profile in the azimuthal direction to make a one-dimensional (1-D) PSF. Next, an encircled energy function (EEF) was derived by integrating the 1-D PSF in the radial direction as follows:Fig. 9. The PSF and EEF were normalized so that the EEF at becomes unity. To evaluate the angular resolution quantitatively, we then derived an HPD. Here, the HPD was defined to be the diameter within which a half of the total x-ray flux () can be accumulated.
The effective area was calculated within a circular region with a radius of 16 mm (4.3′), considering the size of the HXI detector, centered on the peak of the surface brightness. The aperture of the proportional counter in the ISAS/JAXA beamline corresponds to 12 mm (3.4′) in radius. Thus, the effective area below 20 keV was corrected by multiplying by a factor of 1.039, derived from the EEF at 8 keV measured with the CCD camera. The energy dependence of the effective area is shown in Fig. 10. Both the HXT-1 and HXT-2 had sufficiently large effective areas, compared with the requirements: at 30 keV and at 50 keV per mirror module. The effective areas of the HXT-2 were found to be slightly larger than that of the HXT-1. The ratio of the measured effective area to the expected one with an assumption of a constant interfacial roughness of 4.1 Å was found to be 70% to 80% below 40 keV. This ratio gradually declined above 50 keV and then reduced to at 70 keV.
The angular resolutions and effective areas of the HXTs are summarized in Tables 3 and 4, respectively. The typical error of the angular resolution was in HPD. In terms of the imaging quality, the HXT-2 was identical to the HXT-1 within the error. For the effective area, we show errors where only statistical fluctuation of the x-ray counts is taken into account. In Table 3, we also show the diameter within which 90% of the total x-ray flux is contained.
Summary of the angular resolutions of the HXT-1 and 2.
|Angular resolution in HPD (note: typical systematic error is )|
|Diameter of circular region containing 90% fraction of the total flux|
Summary of the effective areas of the HXT-1 and HXT-2.
Values in parentheses represent the corrected effective areas multiplied by a factor of 1.039.
For the HXT-2, we also examined the energy dependence of the effective areas in the 30- to 70-keV band with a finer energy pitch. Since the beam time was limited, we illuminated the x-ray beam only to the five lines of the segment 2 shown in Fig. 11(a). The energy pitch was set to be 1 keV, consistent with the HXI energy resolution. Total effective areas of these five lines are shown in Fig. 11(b). At 37, 44, and 66 keV, the effective areas were measured twice; the measurement reproducibility was found to be 1% to 5%. We also show the ratio of the measured effective area to the expected value at the bottom of Fig. 11(b). The ratio was roughly constant () in the 30- to 40-keV band and then gradually decreased down to at 70 keV, consistent with the trend of the effective area of the whole mirror. Within the systematic uncertainty of this measurement, we cannot find any unexpected artificial structures due to the multilayer design and application.
Stray X-ray Light
Figure 12 shows the stray-light images of the HXT-2 obtained at 12′ and 20′ off-axis angles. The configuration of the stray-light measurement is shown in Fig. 4 of Ref. 26. We rotated the mirror around the -axis with a given off-axis angle and then moved the detector to the location corresponding to the optical axis of the mirror. The pointing scan with this configuration created a distorted focal plane image. The HXT-1 images are already shown in Fig. 15 of Ref. 5, together with those without the precollimator. For the HXT-2, we performed the stray-light measurements at 30 and 60 keV. The stray light is produced by x-rays without normal double reflection. We classify the stray light with the paths of the x-rays inside the mirror.27 The main components of the stray light are a “secondary component” that is reflected only once on the secondary foils and a “backside component” that is reflected on the foil substrates.
As we expected, both components were cleanly removed from the HXI field of view at 20′ off-axis angle. A bright and sharp arc-like pattern represents x-rays that go through a gap between the housing and the innermost (#1) secondary foil without reflection; the inner radius of the housing and the bottom radius of the #1 secondary foil are 56.5 and 59.102 mm, respectively. These x-rays, designated as a direct component, appear within the HXI field of view at 12′ to 21′ off-axis angles. We note that the TS was not attached on the HXT during the on-ground calibration. The inner radius of the TS frame, which is installed on the top of the precollimator, is designed to be 59.85 mm. Hence, we expect that most of the direct component will be removed in orbit.
According to Fig. 3 of Ref. 27, the secondary and backside components correspond to x-rays deflected by to and to , respectively, compared with the normal double-reflection component by . Here, represents the tilt angle of the primary foil relevant to the x-ray reflection. Therefore, the secondary and backside components appear on the opposite sides with respect to the detector center. We calculated the respective effective areas of the secondary and backside components; the extraction regions were indicated with a green rectangle (secondary component) and a cyan rectangle (backside component) in Fig. 12(a). We removed the arc-like patterns to evaluate the effective areas at 20′ off-axis angle. The effective areas are summarized in Table 5. The effective areas within the HXI field of view (secondary and backside components) at 30 keV were found to be at 12′ off-axis angle and at 20′ off-axis angle; these values correspond to and of the on-axis effective area. The ratio at 12′ off-axis angle was well consistent with that estimated from a ray-tracing simulation (see Fig. 4 of Ref. 12).
Summary of the stray light of the HXT-2.
|Effective area of the secondary component ()|
|12′ off axis|
|20′ off axis|
|Effective area of the backside component ()|
|12′ off axis|
|20′ off axis|
On the other hand, the ratio at 20′ off-axis angle was an order of magnitude larger than its estimation from the ray-tracing simulation (). Although the precollimator is designed to sweep out the secondary component at 20′ off-axis angle, misalignment of the precollimator blade positions relative to the foil positions creates a residual secondary component, which results in the effective area of at 30 keV at the 20′ off-axis angle. As for the backside component, the reflectivity and reflected x-ray profiles of the foil substrates above 10 keV were not incorporated correctly in the calibration files for the ray-tracing simulator. Consequently, the backside component may be underestimated in the simulation. The x-ray characteristics of the foil substrates at 20 and 30 keV were measured in the BL20B2 beamline on November 2015. The analysis of the measured data and the accurate evaluation of the contribution of the backside component will be reported elsewhere. Finally, we note that the uncertainty of the flat-field correction ( at the detector center; see Appendix A) may add a systematic error to the estimation of dim x-ray fluxes, such as the backside component.
We also measured the stray light of the HXT-1 at the ISAS/JAXA 27 m beamline. From the pointing scan in the SPring-8 BL20B2 beamline, the x-ray performance on the stray light was obtained for only three sectors. Thus, we focused on the dependence of the stray light on the sectors. To save time, we carried out a one-line raster scan where the x-ray beam of was used to obtain sufficient photon statistics by sacrificing the beam parallelism (the beam divergence becomes 69″). The mirror was rotated around the X-axis stepped by so that the beam illuminates the central part of each sector. The focal plane images at 12′ and 25′ off-axis angles are shown in Fig. 13. At 25′ off-axis angle, the secondary components are expected to vanish within the HXI field of view. However, for some sectors, this component was found to remain at the edge of the detector.
We calculated the effective areas of the secondary and backside components at 8 keV, as shown in Fig. 14. At 12′ off-axis angle, the arc-like direct component, which was blurred by the beam divergence, was overlaid on the backside component. At 25′ off-axis angle, the secondary component dominates the stray light within the HXI field of view. The effective areas of the secondary component at 25′ off-axis angle varied from to . This variation is understood solely by the blade misalignment. If the blade position is shifted radially to the outside of the foil position, the effective height of the blade becomes lower.
We measured the x-ray characteristics of the two HXTs in detail, utilizing the SPring-8 and ISAS/JAXA beamline facilities. We revealed that the HXT performance almost meets the requirements set out to achieve scientific goals. While the angular resolution was 1.9′ at 30 keV and 1.8′ at 50 keV, slightly worse than the requirement of 1.7′, we obtained the effective areas of at 30 keV (13% larger than the requirement) and at 50 keV (49% larger). We also found that the x-ray performance of the HXTs showed a strong energy dependence above 50 keV. In this section, we discuss factors that lead to degradation of the angular resolution and effective areas. Since the characteristics of the HXT-1 and HXT-2 are basically identical, we investigate the factors based on the HXT-2 data only.
Error Budgets of the Image Degradation
We consider here three factors that bring about the image degradation: (1) the conical approximation of the Wolter-I type optics, (2) positioning errors of the foils, and (3) figure errors of the foils. These three factors have been considered so far as the candidates to explain HPDs of various nested thin-foil mirrors (e.g., Ref. 28). According to Table 2 of Ref. 5, the conical approximation contributes to the image spread of 0.3′.
To evaluate the contribution of the other two factors, we inspected the reflected x-ray images obtained from the respective snapshots in the pointing scan. The x-ray image of the ’th snapshot has a line-like shape with a tilting angle of , which corresponds to an azimuthal angle of the ’th pointing position on the mirror aperture illuminated with the x-ray beam (see Fig. 2). Although the positioning error of the foil causes a deviation of the image centroid from the focal position in principle, the figure error is responsible for the image spread. However, we would emphasize here that the incident x-ray beam always illuminated several foils. Therefore, we cannot disentangle the positioning and figure errors of each foil completely with these pointing scan data.
We calculated the position of the centroid of the surface brightness, , for the ’th reflected x-ray image. We divided these centroid positions into three groups based on the radial positions () of each reflected image on the mirror aperture: , , and . The distributions of the centroids at 30 keV are shown in Figs. 15(a)–15(c). The centroids were scattered around the whole mirror focal position of (0′,0′), ranging out to . We note that the centroids cannot be determined precisely for some images taken near the alignment bars or the housing/segment boundaries because of low surface brightness; these structures block a significant fraction of the incident x-ray beam. Such centroids are not plotted in Figs. 15(a)–15(c). We also removed these extremely dim snapshots in the following analysis. The images created from the outer foils tend to be shifted significantly from the whole mirror focal position. A brief comparison of the distributions of the centroid positions between the segments is given in Appendix B. We made a cumulative frequency distribution for the distances of the image centroids from the whole mirror focal position. The distribution curves at 30, 40, 50, and 60 keV are shown in Fig. 16(a). As an indicator of the positioning error, we derived a diameter within which a half of the image centroids (i.e., 783 points) were located. The resultant diameters are summarized in Table 6 as “(2′) positioning error.”
Error budgets of the angular resolution of the HXT-2.
|(1) Conical approximation||0.30′||0.30′||0.30′||0.30′|
|(2′) Positioning error||1.22′||1.09′||1.18′||1.45′|
|(2) Positioning error (flux corrected)||0.81′||0.75′||0.67′||0.67′|
|(3) Figure error||1.63′||1.68′||1.63′||1.56′|
|Root sum of squares of (1) + (2) + (3)||1.84′||1.86′||1.79′||1.72′|
We note that the outer foils have a relatively small contribution to the total x-ray flux of the focal plane image even at 30 keV. Hence, taking into account the difference in the fluxes, we re-estimated image blur caused by the positioning errors by replacing the frequencies with those weighted by a normalized x-ray flux of each image, . Here, represents the x-ray flux of the ’th image. Figure 16(b) shows cumulative distributions of the flux-weighted frequencies. The reflected x-ray images whose centroids are located more than away from the whole mirror focal position were found to contribute little to image blur from the positioning errors. The diameters at which the flux-weighted frequency reaches 0.5 were 0.81′ (30 keV), 0.75′ (40 keV), 0.67′ (50 keV), and 0.67′ (60 keV).
As for the figure errors of the foils, each image was projected onto the radial direction. We estimated a width of this radial intensity profile as an indicator of the figure error. Here, the image width was evaluated by a half-power width (HPW), a range in which 50% of the reflected x-ray flux was accumulated. We made a cumulative distribution of each projected intensity profile and then derived the positions corresponding to 25% and 75% of the total intensity; the distance between these positions was defined as an HPW. The HPW distribution at 30 keV is shown in Fig. 15(d). Each bin in the color map is the position of the incident beam in the pointing scan measurement. The imaging quality, and hence the figure error of the outer foils, was found to be worse than that of the inner ones. We further grouped the snapshots based on their x-ray illuminated positions on the mirror aperture. Using the distances from the mirror center, we first divided the snapshots into nine groups. These groups roughly correspond to the group identification (hereafter, “group ID”) of the multilayer design,5 though some incident x-ray beams of cover two adjacent group IDs. The projected intensity profiles in the same group are shifted so that their centroids are coaligned, added together with the shifted profiles, and then used to derive an HPW. The HPW of each group is summarized in Table 7. The HPWs of the inner foils with group ID 901 to 4 (foil number: #1 to #119) were below , irrespective of incident x-ray energies. For the foils with group ID 5 to 10 (foil number: #120 to #213), the HPWs were energy dependent; the HPWs increase rapidly at 60 keV as the foil radii become large.
HPWs of the reflected x-ray images for each group.
|Group||901 + 902||903 + 904||1 + 2||3 + 4||5||6||7||8||9 + 10|
|No. of pointings||48||87||126||183||153||150||186||234||360|
We also made another grouping for the snapshots; the mirror aperture was divided into 11 regions by a radial range of 60 to 225 mm with a pitch of 15 mm. For each radius, , we collected the snapshots for which the distances of the corresponding pointing positions from the mirror center were less than . The projected intensity profiles of the selected snapshots were again added together with their centroids aligned. The HPWs as a function of the radius on the mirror aperture are shown in Fig. 17. Since the HPW shows a minimum at group ID 903 + 904 or 1 + 2 (Table 7), the cumulative HPWs decrease up to a radius of 105 mm and then gradually increase to 1.63′ (30 keV), 1.68′ (40 keV), 1.63′ (50 keV), and 1.56′ (60 keV). These numbers are also listed in Table 6 as figure errors of the foils.
We calculated the root sum of squares of the figures corresponding to the image blurs caused by the conical approximation, the positioning errors, and the figure errors, assuming that these three factors are independent of each other. The root sums of squares summarized in Table 6 were consistent with the measured HPDs. This result implies that the above three factors are still valid for understanding the imaging quality of the HXTs. However, we note that the weighting by the reflected x-ray fluxes is essential to dealing with the factors quantitatively. We also found that the dominant component that determines the angular resolution was the figure errors of the foils (78% to 82%), while the contributions to the HPDs of the others were 15% to 19% for the positioning errors of the foils and 2% to 3% for the conical approximation. Note that the percentages here are derived on the basis of squares of the figures listed in Table 6.
We investigated the effective areas of each snapshot by comparing them with model calculations. Figure 18(a) shows a color map of the calculated effective areas at 30 keV, (b) that of the measured effective areas, and (c) their ratios. The ratios at 60 keV are also shown in Fig. 18(d). Here, we assumed an interfacial roughness of each group ID to be an averaged value obtained from x-ray measurements for sampled foils; the interfacial roughness was estimated from an angular reflectivity curve at 8 keV that was measured as a quality check of the mirror foils at our x-ray beamline placed at Nagoya University. These roughnesses are summarized in Table 8. We indicated a sample standard deviation as an error in this table. Figure 19 shows a comparison of the effective areas calculated with a constant roughness (, dotted line) and those with the roughnesses of each group listed in Table 8 (dashed line). In the range of 20 to 60 keV, the effective areas calculated with the group-dependent roughnesses were larger by than those with the constant roughness. Above 60 keV, the ratio gradually decreased to be 95%. At least, the measured effective areas were not explained by the interfacial roughnesses actually obtained from the x-ray measurements at 8 keV. As shown in Fig. 10, we need to assume an interfacial roughness of to explain the measured effective areas above 20 keV.
Averaged interfacial Debye–Waller roughness of each group ID for the HXT-2 foils.
|Group||Number of foils||Measured roughness (Å)||Best-fit roughness (Å)|
At 30 keV, the ratios of the measured effective area to the calculated ones were within a radius () of , while those for outer foils reduced to be , except for a band of [see Fig. 18(c)]. This band with a ratio of corresponds to group ID 8. We discuss this issue later and in Appendix C. These ratios at 60 keV decreased to be at ; the effective areas of outer foils were of the calculated ones.
We picked up the snapshots where the housing structures do not interfere with the incident x-ray beam and then calculated the averaged ratios and their standard deviations for each group. The energy dependence of the resultant ratios is shown in Fig. 20. The effective areas gradually decrease relative to our expectation above 50 keV. For the foils of group ID 5, 6, 7, 8, 9, and 10, the decrease in the ratios was found even at 30 keV. We note that the effective areas even of the inner foils were at most 80% of the model calculation.
Furthermore, we tried to estimate the interfacial roughness for each group, which reproduces the averaged effective areas derived from these collected snapshots. We calculated the effective areas for each group at 20, 30, 40, 50, 60, and 70 keV by changing the interfacial roughness from 3 to 12 Å with a pitch of 0.1 Å and then searched the best-fit roughness that yields the minimum . The best-fit roughnesses are given in Table 8. To reduce the effective area, the best-fit roughnesses tend to be larger than those estimated from the angular reflectivity curves at 8 keV, though the roughnesses for group ID 901, 902, 9, and 10 were not constrained with an accuracy of due to large errors on the averaged effective areas. We also calculated the energy dependence of the effective areas when adopting a combination of these best-fit interfacial roughnesses in Fig. 19. The curve reproduced well the measured effective areas in the 30- to 70-keV band. However, the effective areas below 20 keV were still overestimated by at 4.5 and 8 keV. This result indicates that the energy dependence of the effective areas cannot be explained solely by the group-dependent interfacial roughnesses of the multilayers and that we should take into account other causes to reduce the effective areas below 20 keV.
We assumed here a single interfacial roughness for each group ID, independent of incident x-ray energies. However, the interfacial roughnesses are actually energy dependent (see Appendix C) since the penetration lengths to multilayer are different by x-ray energies. Therefore, the x-ray reflectivity should be measured at multiple energies for flight-quality foils to inspect the interfacial roughnesses of each stacked layer. The accurate modeling of the x-ray reflectivity over a wide energy band is essential to minimizing the uncertainty of the roughness contribution to the energy-dependent decrease in the effective areas.
These ratios of the measured effective areas to the calculated ones (hereafter throughput) have been considered to be reduced by geometrical effects, e.g., blocking or scattering of a fraction of incident x-rays due to positioning errors and figure errors of the tightly nested foils. Another possibility is the degradation of the reflectivity that can be caused by imperfection in the multilayer coating.
Since the snapshots in the pointing scan made overlaps of multiple foils, we further took defocused images at 30 keV with a flat panel C7942CA-22 (CsI scintillator + CMOS image sensor) placed just behind the mirror. This flat panel has a pixel size of . The distance between the flat panel and the HXT bottom was set to be 115 mm. We made a beam by the four-jaw slit and then put a stainless-steel mask with 6-mm thickness between the slit and the mirror to make a fan-shaped incident beam with a central angle of 1.714 deg. The HXT was rotated around the -axis with an angular pitch of 1.714 deg; 210 defocused images were obtained to cover the entire 360 deg. The start angle corresponds to the boundary between the segments 1 and 3. We designated this start angle as a sequence number 1 and then increased the number with the rotation. Since the radial width of the incident beam was only 20 mm, we moved the mirror in the -direction to change the radial range that the x-ray beam illuminates. We set a total of nine radial ranges to cover the full aperture of the mirror, designated as R60, R80, R100, R120, R140, R160, R180, R200, and R220. Figure 21 shows an example of the defocused images, which covers radial ranges of to 80 mm and to 210 mm.
We made an intensity profile projected on the -axis from each defocused image. The incident x-ray beam was found to have nonuniform spatial distribution of the surface brightnesses in the -direction. The projection width was set to 0.8 mm (16 pixel), indicated with green boxes in Fig. 21, to use only an area with a relatively flat distribution of the surface brightnesses. We note that this width corresponds to an azimuthal angle of 0.76 deg and 0.20 deg for the innermost and outermost foils, respectively. As a result, all of the 210 defocused images cover an area of 44% and 12% of the innermost and outermost foils, respectively. Figure 22 shows an example of intensity profiles. The intensity profiles were normalized with the intensity of the direct beam.
To evaluate the data, we made a model curve calculated from a step function of , multiplied with the square of the reflectivity, . The step function is defined to be 1 for , otherwise 0, where and represent the inner radii of the top of the ’th and ’th primary foils, respectively, and the 0.22 mm is the thickness of the foil substrate. is the incident angle of the ’th primary foil. The model curve draws 213 trapezoids in the intensity profile. Before applying this trapezoidal model to the data, we need to take the beam divergence into account. In doing this, we utilized a sharp edge at in the R60 intensity profile, which was caused by shadowing of the incident beam by the inner wall of the mirror housing. We fit this shadowing profile with a one-sided double Gaussian function. The best-fit model was used to smooth the trapezoidal model curve. Finally, we shifted the smoothed trapezoidal model curve in the radial direction to minimize residuals between the valleys of the data and model.
The measured intensity profile has a kind of sawtooth shape, which is much narrower than the smoothed trapezoidal one indicated with dotted curves in Fig. 22. We calculated the areas of each smoothed trapezoid, which represent the reflected x-ray fluxes of the corresponding foil pair. The measured x-ray fluxes were also obtained in the same manner; the ’th area within a range of was calculated from the measured intensity profile. Figure 23 shows the ratios of the measured x-ray fluxes to the calculated ones for each foil pair. Here, the ratios were averaged over the total 210 sequences. We note that the ratio for each foil pair shows a large scatter with a fluctuation amplitude of . While the ratios of the foil pairs below #100 are typically 70% to 80%, those of the foil pairs with larger number gradually decrease to be . As already shown in the color map of Fig. 18(c), the ratios of group ID 8 exceed unity. In addition, the ratios of several foil pairs around #192 and #213 exhibit a relatively high value of .
To reproduce this trend of the ratios as well as the sawtooth shape of the measured intensity profiles, we first examined the geometrical effect due to the positioning errors of the foils. We tried a simple ray-tracing simulation to investigate the effect of a radial shift of the foil edges on the intensity profile itself. Figure 24 shows intensity profiles of the #208 foil pair, where the top edge of the primary foil and the bottom edge of the secondary foil were shifted in the radial direction by , , and . All of the simulated profiles had a trapezoidal shape, inconsistent with the measured profile. Moreover, while the intensity of the shifted profiles changed with the amount of the shift, their widths appeared to be constant. We also note that the radial positions of the alignment bars were tuned with an accuracy of during the mirror assembly process so that the focusing x-ray image becomes axisymmetric and makes a single peak at 30 keV.5 Thus, even the amount of is an unrealistic radial shift due to positioning errors.
However, even if the alignment bars were tuned accurately, the edges of each foil can move in the corresponding groove. The image blur of due to the positioning errors (see Table 6) was indeed larger than 0.14′ estimated as the image blur caused by misalignment of the alignment bars ().5 Thus, we inspected a random shift of the radii of the top/bottom edges of the foils. We assumed a Gaussian distribution with an average of and a standard deviation of . If both the top and bottom edges are shifted by in the opposite direction, the foil tilts by 0.5′ from its designed configuration. We introduced this random shift to all of the primary and secondary foils and then performed calculations of the effective areas 5000 times in the range of 20 to 70 keV. For each x-ray energy, the 90% confidence interval of the effective area was derived, as shown in Fig. 25. This figure indicates that the positioning errors of the foils due to the random shifts in the grooves change the effective areas by 7% at most. Hence, the energy dependence of the measured effective areas was not fully explained by the changes of the incident angles caused by the random foil positioning errors.
In terms of the sawtooth shape of the intensity profile, we probably need to introduce the effect of the figure errors of the foils. Judging from the HPWs in Table 7, there may be a correlation between the energy dependences of the effective areas and the HPWs for each group ID, indicating that the figure errors of the foils are key factors to explaining the degradation of the effective areas at higher energies. The profiles shown in Fig. 22 represent intensity profiles with double reflection for each foil pair. Since the reflected profiles of the primary and secondary foils are convolved with each other, we cannot estimate the slopes of the foils from these data only. Therefore, for the future progress in understanding the performance of hard x-ray mirrors, we suggest here defocused x-ray measurements with (1) different x-ray energies (e.g., 60 keV) and (2) different defocused positions (e.g., 200 mm from the mirror). The defocused measurement during the mirror assembly process, where either primary or secondary foils are installed in the mirror housing, would be also useful to evaluating the effect of the figure errors with a single reflection.
We completed the on-ground measurements of the two HXTs in November 2014 using the SPring-8 BL20B2 beamline and the ISAS/JAXA 27 m beamline. The x-ray performance of the two HXTs was nearly identical. The angular resolution (HPD) was 1.9′ (HXT-1) and 2.1′ (HXT-2) at 8 keV, 1.9′ at 30 keV, and 1.8′ at 50 keV, close to the requirement of 1.7′ (see Table 1). The effective area was obtained to be (HXT-1) and (HXT-2) at 30 keV and (HXT-1) (HXT-2) at 50 keV; the effective areas of the HXT-2 were slightly larger than those of the HXT-1. In terms of the effective area, the HXTs fully meet the requirements of (30 keV) and (50 keV). The effective area at 8 keV was found to be per HXT module. The detailed energy dependence of the effective area for a limited aperture was measured in the 30 to 70 keV with a pitch of 1 keV. Compared with the model calculation assuming an interfacial roughness of , no unexpected artificial structure due to the multilayer design was found. These results will be used to tune basic parameters of the calibration files for the ray-tracing simulator that are used to make response functions.
Furthermore, we measured the angular dependence of the effective areas at 8, 30, and 50 keV. From the vignetting curves, the fields of view, defined to be the full-width half maximum of the curves, were estimated to be 9.7′ at 8 keV, 7.7′ at 30 keV, and at 50 keV, slightly larger than the expectation from the ray-tracing simulations. Stray-light measurements were also carried out. We verified the reduction of the stray light from the focal plane images. We also found that the fluxes of the stray light were different among the sectors. This difference is probably caused by the sector-dependent misalignment between the foils and the precollimator’s blades. For the quantitative evaluation of the residual stray-light fluxes, we need a detailed study of the x-ray characteristics of the foil substrates above 10 keV.
We examined the focal plane images taken from the spot illumination in more detail to figure out the key factors that determine the angular resolution and the effective areas. The HPD can be well explained by the root sum of squares of the image spreads by (1) the conical approximation of the Wolter-I optics, (2) positioning errors of the foils, and (3) figure errors of the foils; the dominant factor that determines the angular resolution was found to be the figure errors. We also found that the flux contributions to the total x-ray image should be taken into consideration for the accurate evaluation of the positioning and figure errors. For example, although the figure errors of the outer foils exceeded 3′ at 60 keV, their contribution to the figure errors of the whole mirror was 1.56′.
The ratios of the measured effective areas to those derived from the model calculation were found to be below 40 keV and to gradually decrease to at 70 keV. We then investigated the local effective areas using the pointing scan data. The contributions to the effective areas from the outer foils were found to be rapidly reduced with increasing x-ray energies. This energy dependence was neither attributed to the error of the multilayer design (Sec. 4.2) nor explained by a change of the incident angles of the foils, which was caused with the positioning errors of the foils; the change of the effective areas due to the positioning errors was at most 7% (Sec. 4.3). We also obtained the defocused images at 30 keV to examine the flux contribution of each foil pair. The intensity profile of the defocused images had a sawtooth-like shape, implying that the figure errors of the foils affect the reflected x-ray fluxes. Although there is little information on the effect of the figure errors on the reduction in the effective areas, the incident x-rays on the outer foils may be largely scattered at higher energies on the basis of the image spread due to the figure errors.
Flat Field of the Image Intensifier
The sensitivity of the image intensifier (I.I.) was known to be nonuniform over the () field of view; the sensitivity of the central part was degraded, compared with its surroundings, due to frequent illumination of an intense x-ray beam with a specific size. Hence, we applied a sensitivity correction to each focal plane image, to evaluate the exact x-ray surface brightness. We made sensitivity maps of the I.I. before starting the x-ray measurements of the HXT-1 and HXT-2.
First, we made a strip of the incident x-ray beam with a height of 1 mm, within which the beam was assumed to be uniform in the -direction. Meanwhile, the width of the beam was set to be sufficiently larger than the detector width (77.2 mm). The strip beam was scanned in the -direction to illuminate the whole detector [see Fig. 26(a)]. We repeated this measurement, took 11 frames and then summed them together. Since the time variability of the intensity of the incident x-ray beam was only , we did not apply any intensity correction to each frame. The summed image showed a ghost component near the center, caused by optics incorporated in the I.I.; multiple reflection between the optical lens and the CCD chip may create the ghost image [Fig. 26(b)]. By illuminating the I.I. with the strip beam at the positions from the detector center, we took the ghost image only. The spatial distribution of the surface brightness of the ghost component was modeled with a two-dimensional Lorentz function. We multiplied the best-fit model by a factor and then subtracted it from the summed frame. The factor was determined so that the sensitivity over the detector except around the center becomes uniform after the subtraction.
Finally, we investigated the intensity profile of the strip beam along the -direction [Fig. 26(c)]. We again illuminated the strip beam to the I.I. and then moved the detector in the -direction with a 1-mm pitch. The beam intensity accumulated from a specific region was monitored to reconstruct the intensity profile. Three intensity profiles were made by illuminating the strip beam at , from the detector center. We made a smooth function by the spline interpolation of the averaged profiles. The flat-field image corrected for the intensity profile of the strip beam is shown in Fig. 27. We note that this flat-field image does not reproduce completely the degraded sensitivity of the central part; an uncertainty of still remains at the center. Away from the center, the uncertainty of the sensitivity map was .
As described in Ref. 5, we tuned the radial positions of the alignment bars so that the focusing x-ray spots of the beam at 30 keV could be adjusted to the center of the detector. Thus, the focal length of the HXT becomes exactly 12 m in the ideal case. After the tuning of the bar positions, however, we found that the standard deviation of the centroid positions of each reflected x-ray image against the detector center was .5 This deviation means that the actual focal length may be different from 12 m by . We assumed here a radius of 100 mm, where the contribution of the reflected x-ray intensity to the on-axis effective area at 30 keV became maximum.
No x-ray measurements to examine the actual focal length of the HXTs were carried out. However, we measured the focal lengths of each segment utilizing an optical parallel beam to inspect the HXTs during the environmental tests. For the HXT-2, the focal lengths before the on-ground calibration were 11.887 m (segment 1), 12.015 m (segment 2), and 11.983 m (segment 3). On the other hand, from the distribution of the centroid positions of the reflected x-ray images [Figs. 15(a)–15(c)], simple averages of the centroid positions were calculated to be (segment 1), (segment 2), and (segment 3). Although their variances are large, relative to the positioning errors, the averaged position of the segment 1 is far away from the detector center by ; it corresponds to the difference in the focal length of . Thus, the averaged positions of the image centroids are considered to be a good indicator for the relative difference in the focal lengths among the segments. We also note that the angular resolution of the HXTs is insensitive to their focal length, at least within a range of .
Reflectivity of the Group ID 8 Foil
At the SPring-8 BL20B2 beamline, we investigated the cause of the reflectivity enhancement of the group ID 8 foils obtained at 30 and 60 keV. We made a multilayer used in the group ID 8 foils on a float glass and then replicated it onto another float glass, similar to the replication process of the foils. This float glass was set on a -axis translation stage that was placed on a rotation stage. We illuminated the float glass by a beam and then took images of the reflected x-rays with the scintillator + CCD camera. The distance between the float glass and the scintillator was set to be 2500 mm.
The reflected x-ray images were corrected for the sensitivity map of the detector. We also applied the correction of the time variability of the direct beam intensity and the dark level of the detector. Moreover, we made intensity profiles projected on the -axis from the reflected x-ray images. A set of the peak positions obtained from the profiles was used to estimate an angle offset of the float glass. The reflectivity curves at 30 and 60 keV are shown in Fig. 28.
We also superposed the best-fit models in Fig. 28. The interfacial roughnesses of Pt layers were found to become large with an increase in the x-ray energy; the roughness of 5 Å at 8 keV increased to 6 Å at 30 keV or 9 Å at 60 keV. We note here that the Nevot–Croce model, instead of the Debye–Waller model, was used to estimate the interfacial roughnesses more accurately. It was empirically found that the roughness derived from the Nevot–Croce model is times larger than that from the Debye–Waller model. The increase in the roughnesses indicates that the different parts of the stacked layers were examined by the different x-ray energies; the measurement at 8 keV allows us to estimate the roughness only for a few topmost layers.
We note that the group ID 8 foils cover an incident angle of 0.219 deg to 0.241 deg (see Table 3 of Ref. 5), corresponding to the angular range where the reflectivity at 30 keV changes rapidly by . This result suggests that a slight change of the incident angle induced by the foil displacement easily causes the enhancement of the reflectivity, leading to the increase in the effective area. We also note that the increase in the interfacial roughness affects little the reflectivity curve in this angular range (0.219 deg to 0.241 deg; see Fig. 28). For example, at 30 keV, the averaged ratio of the square of the reflectivity with to that with was 97%. Therefore, we should take into consideration the other causes, other than the interfacial roughness, to reproduce the rapid decrease in the effective areas of the outer foils at higher energies (see Figs. 18 and 20).
We would like to express our gratitude to the anonymous reviewers for their fruitful comments to improve our article. We also would like to thank Dr. Tahir Yaqoob (NASA/GSFC) for his careful English check and correction to our article. We appreciate the useful comments and support of the full-time engineers and part-time workers in Nagoya University, who dedicated their efforts to the mass production of the HXT foils. We are grateful for the technical/financial supports from the Japan Synchrotron Radiation Research Institute (JASRI)/SPring-8. The x-ray measurements at the SPring-8 BL20B2 beamline were performed with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal Nos. 2009A0088, 2009B0088, 2010A0088, 2010B0088, 2011A0088, 2011B0088, 2012A0088, 2012B0088, 2013A0088, and 2013B0088). This work was supported by the Grant-in-Aid for Scientific Research (A), JSPS KAKENHI Grant No. 15H02070.
Hideyuki Mori is an assistant research scientist at the Center for Research and Exploration in Space Science and Technology (CRESST II). He is a member of SPIE.