Multiplexing lobster-eye optics: a concept for wide-field X-ray monitoring
Toru Tamagawa, Keisuke Uchiyama, Ryota Otsubo, Tatsuya Yuasa, Yuanhui Zhou, Tatehiro Mihara, Yuichiro Ezoe, Masaki Numazawa, Daiki Ishi, Aoto Fukushima, Hikaru Suzuki, Tomoki Uchino, Sae Sakuta, Kumi Ishikawa, Teruaki Enoto, Takanori Sakamoto
MMultiplexing lobster-eye optics: a concept for wide-field X-raymonitoring
Toru Tamagawa a,b,c* , Keisuke Uchiyama b,c , Ryota Otsubo d , Tatsuya Yuasa d , YuanhuiZhou b,c , Tatehiro Mihara a,b , Yuichiro Ezoe d , Masaki Numazawa d , Daiki Ishi d , AotoFukushima d , Hikaru Suzuki d , Tomoki Uchino d , Sae Sakuta d , Kumi Ishikawa e , TeruakiEnoto a , Takanori Sakamoto f a RIKEN Cluster for Pioneering Research, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan b RIKEN Nishina Center, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan c Tokyo University of Science, 1-3 Kagurazaka, Shinjuku, Tokyo 162-8601, Japan d Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji, Tokyo 192-0397, Japan. e Japan Aerospace Exploration Agency, Institute of Space and Astronautical Science, 3-1-1 Yoshino-dai, Chuo-ku,Sagamihara, Kanagawa 252-5210, Japan f Aoyama Gakuin University, 5-10-1 Fuchinobe, Chuoku, Sagamihara, Kanagawa 252-5258, Japan
Abstract.
We propose a concept of multiplexing lobster-eye (MuLE) optics to achieve significant reductions inthe number of focal plane imagers in lobster-eye (LE) wide-field X-ray monitors. In the MuLE configuration, an LEmirror is divided into several segments and the X-rays reflected on each of these segments are focused on a singleimage sensor in a multiplexed configuration. If each LE segment assumes a different rotation angle, the azimuthalrotation angle of a cross-like image reconstructed from a point source by the LE optics identifies the specific segmentthat focuses the X-rays on the imager. With a focal length of 30 cm and LE segments with areas of 10 ×
10 cm , ∼ ×
10 cm ). A raytracing simulation was performed to evaluate the nine-segment MuLE configuration. The simulation showed that theflux (0.5 to 2 keV) associated with the 5 σ detection limit was ∼ × − erg cm − s − (10 mCrab) for a transientwith a duration of 100 s. The simulation also showed that the direction of the transient for flux in the range of 14 to17 mCrab at 0.6 keV was determined correctly with 99.7% confidence limit. We conclude that the MuLE configurationcan become an effective on-board device for small satellites for future X-ray wide-field transient monitoring. Keywords:
X-ray all-sky monitoring, transient monitor, lobster-eye optics, multiplexing lobster-eye, small satellite. * Toru Tamagawa, [email protected]
Wide-field X-ray monitors have been proven to be indispensable devices in time-domain astron-omy in recent years. The precise and immediate localization of transient phenomena is criticalfor the revelation of their origin. For example, quick localization of gamma-ray bursts (GRBs) re-vealed the origin of GRBs with long durations as collapsers. Gravitational waves from a neutronstar merger were detected in 2017 and demarcated the onset of multimessenger astronomy. In2018, follow-up observations were carried out for the neutrino burst detected by IceCube, and theorigin of this event was localized to an active galactic nucleus. In 2021, the large synoptic surveytelescope will begin its observations and will generate several million alerts per night. Identifyingthe high-energy counterparts of these visible transients is important for elucidating their origins.Correspondingly, in multimessenger astronomy, the use of a device that constantly monitors theuniverse with a wide field-of-view (FoV) in the X-ray energy band is essential.Coded masks are used for wide FoV missions such as
INTEGRAL , Swift/BAT , HETE/WXM , and BeppoSAX/WFC , but in principle it is difficult to increase the sensitivity because of the1 a r X i v : . [ a s t r o - ph . I M ] J u l nterference caused by the diffuse cosmic X-ray background (CXB). The all-sky monitors on-board RXTE and MAXI improved the detection sensitivities by narrowing the FoV with pinholecamera or slit techniques, and yielded excellent performance in the observation of faint X-raysources. To compensate for the improved sensitivity, the sky coverage of a moment was restrictedto a few % of the entire sky.Lobster-eye (LE) optics represents the best possible observation equipment for missions thatrequire a wide FoV and increased sensitivity. The LE optics reduces the influence of the CXB byfocusing, and concurrently securing a broad FoV. Several X-ray astronomical satellite missions,such as Einstein Probe , ISS-Lobster , and HiZ-GUNDAM employ the LE optics. A disadvan-tage of the LE optic is the necessity for large-sized imagers at the focal plane. For example, theall-sky monitor mission LOBSTER requires a detector area spanning 5000 cm to cover ∼ In this study, we describe the design, feasibility, and performance evaluation of anewly proposed idea of LE optics to reduce the number of imagers.
The LE mirror consists of many square, hollow cells that operate as X-ray reflectors tiled on acurved sphere with a radius R , as shown in Fig. 1. X-rays that originate from a point source arereflected twice on the adjacent walls of a square hollow cell (Fig. 1b) and are focused on a point onthe focal plane with a radius of R /2. When the incident angle of the X-rays is different, the X-raysare focused on another location on the focal plane. In combination with the image sensors placed atthe focal plane, the LE optics realizes X-ray imaging with wide FoV that is not achievable with anyother standard X-ray mirror optics. As shown in Fig. 1b, since the X-rays reflected only once in theX (Y) surface of a cell are focused in the X det (Y det ) direction but not in the Y det (X det ) direction,the focus should be a line along Y det (X det ). Thus, those photons are focused on cross-like armfoci. To cover the entire FoV of the LE mirror, which is the opening angle of an LE mirror segmentas described in Appendix A, large image sensors covering a 1/4 size of the area of the LE mirrorare required at the focal plane. However, imagers with large areas are sometimes unsuitable for asmall satellite mission because they consume non-negligible satellite resources, such as electricalpower, computer power, and data downlink bandwidth, and may cause cooling problems.To overcome these disadvantages, we propose a new configuration in which the LE mirror isdivided in several segments and the X-rays reflected on each segment of the mirror are focused ona single, small image sensor, as shown in Fig. 2. If we define the opening angle of an LE segmentas 2 θ , the LE segment ID20 in Fig. 2, which is 4 θ away from the central segment (ID00), can bemoved right next to the ID00 segment. We refer to this configuration in this study as ”multiplexinglobster-eye (MuLE)” optics. To specify a LE segment, we use the notation ID n x n y , where n x and n y are indices used to represent the distance of the segment nθ away from ID00 in the x and ydirections, respectively. Negative integers are represented with a bar. For example, ¯ n implies − n .The similar configuration was adopted by the ABRIXAS mission in which one CCD camera wasshared by seven X-ray mirrors. Their design was to drop different FoVs to different areas of theimager, but in our concept different FoVs are dropped to the same area of an imager.How can we distinguish two stellar objects focused by different LE segments on one imager?As shown in Fig. 1b, a point source focused by an LE segment shows a cross-like response on theimager. The azimuthal rotation angle of the cross-like arm foci on the imager is exactly the sameas that of the square hollow cells of the LE segment around the central optical axis of the segment.2 -ray mirror cells Focal plane (a) (b) RR/2 lobster_eye_v08.pdf X cell Y c e ll -X +X +Y-Y X d e t Y d e t Fig 1 (a) Schematics of the LE optics showing the X-ray mirror cells mounted on a curved spherical surface with aradius R and focal plane detectors at a focal length of R /2. X-rays from different positions in the sky are focused ondifferent locations on the focal plane. (b) A square, hollow cell and the path of an X-ray scattered on different planes+Y and -X. The X-ray photon is focused on the center of a cross-like image generated by the LE optics on a focalplane detector. The photons reflected on the mirror once are focused on the cross-like arm foci. By giving each segment a different azimuthal rotation angle, point sources focused by differentmirror segments form cross-like arm foci with different azimuthal rotation angles.We estimated the total FoV covered by the MuLE optics. The half angle θ of the FoV of eachLE segment was defined as θ = sin − ( L/ R ) , where each LE segment had an area of L × L . Ifwe consider specific values R = 60 cm and L = 10 cm, the FoV of each segment becomes . ◦ × . ◦ . One of possible configurations of the MuLE optics consists of nine tiled segments, as shownin Fig. 3, in which an azimuthal rotation angle of each LE segment increments 10 ◦ from 0 ◦ to 80 ◦ .The numbers φ φ
80 in Fig. 3a indicate the azimuthal rotation angles of the LE segment cellsaround the optical axis. It is not difficult to manufacture such mirrors with current technology.As observed from Fig. 2, the FoV covered by ID20 is not the continuous tiling of ID00 but atiling configuration at every other position in the sky coordinate system. When four units of thenine-segment MuLE are used, a sky area of 57.4 ◦ × ◦ can be covered, as shown in Fig. 4. Eachunit is named A to D, and an imager is installed directly under ID00 of each unit. The four ID00sof units A to D are installed offset from each other by 2 θ in the x and y directions. Since eachsegment of the nine-segment MuLE covers the FoV every 4 θ (Fig. 2), it is possible to continuouslycover the FoV with four units of nine-segment MuLE. Accordingly, we can achieve an FoV of ∼ ×
10 cm ).3 =R/2 θLE mirror segmentImager 4θ RID00 I D leoptics_v08.pdf xz Segment S e g m e n t Fig 2
Conceptual design of the MuLE optics to achieve reductions in the number of imagers. A single imager isshared between ID00 and ID20 LE mirror segments.
To evaluate the performance of the MuLE optics, including the surface roughness of the mirrors,misalignment of the LE mirror cells, and a realistic detector configuration, we performed a raytracing simulation by modifying a previously built simulator. The values of the surface roughnessand the mirror cell misalignment were taken from our past mirror fabrication. In this study, wetreated only one parameter set because we aimed to evaluate the working principle of the MuLEoptics. The optimization of the parameters will be discussed in our next publication.
The simulation was performed with the nine-segment MuLE configuration shown in Fig. 3. OneLE segment has a size of 10 ×
10 cm . Given that the support structures of the LE segments arenecessary in a realistic design, a 0.5-cm margin was added around each LE segment. Thus, thegeometrical area of each LE segment becomes 9 × . The nine LE segments are tiled on aspherical surface with a radius R = 60 cm. The azimuthal rotation angle of each LE segment isincremented by 10 ◦ from 0 ◦ (central one) to 80 ◦ , as shown in Fig. 3.Recently, some X-ray mirrors have been produced with a silicon–microelectromechanical sys-tems (Si–MEMS) technology that is a precise and a less expensive technique applicable to the LE4 e_module_config_v09.pdf ID22 -φ60 ID02 -φ70 ID22 -φ80ID20 -φ50 ID00 -φ00 ID20 -φ10ID22 -φ40 ID02 -φ30 ID22 -φ20 X LE Y L E (a) (b) Fig 3 (a) The configuration of nine-segment MuLE optics. The azimuthal rotation angle of the square, hollow cellsof each segment is shifted by 10 ◦ . The numbers begin with φ after the use of the LE segment ID to represent theazimuthal rotation angles around the optical axes of the LE segments. The cell size of each segment is exaggerated.(b) Three-dimensional modeling of the nine-segment MuLE optics. A 22-φ40 A 02-φ30 A 22-φ20A 20-φ50 A 00-φ00 A 20-φ10A 22-φ60 A 02-φ70 A 22-φ80B 22-φ40 B 02-φ30 B 22-φ20B 20-φ50 B 00-φ00 B 20-φ10C 22-φ40 C 02-φ30 C 22-φ20D 22-φ40 D 02-φ30 D 22-φ20C 20-φ50 C 00-φ00 C 20-φ10D 20-φ50 D 00-φ00 D 20-φ10C 22-φ60 C 02-φ70 C 22-φ80D 22-φ60 D 02-φ70 D 22-φ80B 22-φ60 B 02-φ70 B 22-φ80 fieldofview_v05.pdf
12 sin -1 (L/2R)Θ X Θ Y s i n - ( L / R ) Fig 4
FoV covered with four units of the nine-segment MuLE optics. A to D denote the unit numbers. For R = 60 cmand L = 10 cm, the angular span of 57.4 ◦ × ◦ ( ∼ µ m, and the pore size was 20 × µ m . Since the spacing between adjacent pores was40 µ m, the open fraction of the aperture was 25%. To keep the structural strength of the Si–MEMSmirror, radial spokes with widths of 300 µ m were added every 15 ◦ . This reduced the aperture ratioto 21%. Compared with the standard LE mirror made of glass material, the thickness is aboutone-third, but the other properties such as the point spread function are comparable.In the simulations, X-rays originating from the nine LE segments were captured by a 4 k × × (i.e., with a 15- µ m pixel size) centeredat the focal point of f = 30 cm. The state-of-art complementary metal-oxide semiconductor(CMOS) technology allows us to use low-noise pixel imagers without cooling. GPixel’s CMOSsensors represent these types of devices. The 15- µ m pixel size corresponds to the arc lengthof 10 arcsec in the sky coordinate system. It is small enough compared with the imaging qualityof the Si–MEMS LE optics. The detailed values of parameters for the ray tracing simulation aresummarized in Table 1. Table 1
Parameters of the ray tracing simulation
Parameter ValueScan energy ( E ) 0.5 to 3.5 keV (0.5 keV step)Scan angle of photons ( Θ x , Θ y ) * ◦ (2 ◦ step)Thickness of lobster-eye (LE) mirror ( (cid:96) ) 300 µ mRadius of LE sphere ( R ) 60 cmFocal length of LE optics ( f ) 30 cmOpen fraction of cells ( η ) † w × w ) 20 × µ m LE mirror segment size ( L × L ) 10 ×
10 cm LE mirror effective area ( L e × L e ) 9 × Mirror coating material PtMirror surface roughness ‡ ‡
10 arcmin (FWHM)Imager size § × * Scan angles are measured from the center of the field-of-view of eachLE segment. † Shadows induced by the radial spokes are included. ‡ These values were obtained from our Si–MEMS manufacturing expe-rience. § × µ m). First, we simulated the image response of a point source focused by the ID00- φ
00 LE segment.Figure 5a shows the image response of a point source with an incident angle of Θ x = Θ y = 0 ◦ withrespect to the central optical axis of the ID00- φ
00 segment. Approximately 15% of the detectedphotons at 0.6 keV were scattered twice on the adjacent walls of the LE cells and focused at thecenter of the imager (marked as ”Focus” in the figure). Approximately 48% of the photons were6 det (cm) X det (cm) Y d e t ( c m ) Y d e t ( c m ) (a) (b)(c) (d) 10 C o un t s / b i n pointsource_images_v10.pdf Focus ArmX A r m Y NoRef - - - - Fig 5 (a) Simulated image of a point source focused by ID00- φ
00 with an incident photon angle of Θ x = Θ y = 0 ◦ .See text for detail on ”Focus” and ”ArmX/Y”. The dashed lines show the boundary beyond which there are notnon-reflected (NoRef) photons. The red rectangle shows the size of the imager. (b) ID00- φ
00 with Θ x = 4 ◦ and Θ y = 0 ◦ . The dashed line shows the image boundary produced by the edge of the LE segment. (c) ID20- φ
10 with Θ x = Θ y = 0 ◦ . (d) ID22- φ
20 with Θ x = Θ y = 0 ◦ . The color bar shows the counts-per-bin on a logarithmic scale.All of the images were reconstructed with ∼ scattered once on the cells and concentrated in the cross-like arm foci (these are marked as ”ArmX”and ”ArmY”). The remaining 37% of the photons were dropped through the cells directly to theimager (these are marked as ”NoRef”). The boundary limit angle beyond which NoRef photonsdo not exist is defined by θ lim = tan − ( w/(cid:96) ) = 3 . ◦ , and corresponds to 4 cm ( = R sin θ lim ) onthe imager.Second, we considered the point source with an incident angle of Θ x = 4 ◦ and Θ y = 0 ◦ withrespect to the central optical axis of ID00. Figure 5b shows the image of the source clearly shiftedto the right compared with Fig. 5a. Only half of the image was detected in the X-axis direction, butit was sufficiently detected even at the edge of the FoV. In realistic configurations used in X-rayastronomy, the missing half of the X-ray images could be detected by another MuLE unit giventhat the FoVs are tiled withoug gaps, as shown in Fig. 4, i.e., the reduction of the effective areacan be almost mitigated. The boundary created by the edge of the LE segment is clearly seen inFig. 5b at X det = 2 . for Θ x = 4 ◦ . See Appendix A for a detailed description of the edge of7he LE segment.Finally, we simulated the point source images focused by the ID20 and ID22 segments. Fig-ure 5c shows the image focused by the ID20- φ
10 segment. The image response was similar to thatof ID00 but was rotated 10 ◦ as the LE segment rotated. Figure 5d shows the image focused by theID22- φ
20 segment. The cross-like images in both the ID20 and ID22 segments were clearly seen.This implied that the images from any LE segment could be detected.As expected, defocus aberration was observed at the edge of the CMOS image sensor for theID20 and ID22 segments, given that the focal plane was tilted in these segments. The worst case ofthe defocus aberration appeared at the diagonal edge of the CMOS imager for the ID22 segment.At that point, the focal length was ∼ f = 30 cm . The defocus corresponds to ± The mirror effective areas were also derived from the ray tracing simulation. Figures 6a–6d showthe effective areas of ID00 as a function of the incident photon angle measured from the opticalaxis of the LE segment for 0.5, 1.0, 2.0, and 3.5 keV, respectively. In this calculation, the size ofthe CMOS sensitive area was taken into account, but the quantum efficiency of the imager was notsince the efficiency is almost 100% in this energy band. The simulations were conducted basedon discrete calculations within the angle range of Θ x at 2 ◦ steps. The reason for including NoRefin the figures of the effective areas was that the LE optics had two functions: a focusing mirror(ArmX/Y and Focus) and a collimator (NoRef). Since the density of X-ray objects in the sky issparse, if no other object is in the FoV, NoRef is identified as X-rays from the target object.The curved lines shown in Figs. 6 were analytically calculated effective area in combinationwith the mirror reflectivity. The detailed procedure of the analytic calculation is summarized inAppendix B. The discontinuity marked (i) in Fig. 6a shows the angle where the Focus is shiftedoff the edge of the CMOS. The effective area for ID20 and ID22 at 0.5 keV are shown in Figs. 7aand 7b respectively. Given that the difference between ID20/22 and ID00 is originated only inthe tilt angle of the X-ray images, the curves of the effective area look very similar to each other.To clarify the characteristics of the nine-segment MuLE optics only, the vignetting is shown inAppendix C.The effective areas of the mirror as a function of the incident photon energy for the ID00 seg-ment with an incident angle of Θ x = Θ y = 0 ◦ are shown in Fig. 8. The ray tracing simulation wasperformed for different energies at every 0.5 keV from 0.5 to 3.5 keV. The curves of the effectiveareas derived from the analytic calculation are also shown in the figure. While the effective area ofNoRef was flat, the effective areas of ArmX/Y and Focus dropped rapidly as the energy increased.The effective area of ArmX+Y was somewhat larger than that of NoRef below 1 keV. The source detection limit was determined by the signal-to-noise ratio of the X-ray photons onthe imager. In the MuLE optics, the most dominant noise is the diffuse CXB. Fig. 9 shows the5 σ detection limits for Focus, Focus+ArmX/Y, and total (Focus+ArmX/Y+NoRef) when a point8 ncident photon angle (deg)0 2 4 6 8 10 0 2 4 6 8 10 E ff e c t i v e a r e a ( c m ) (a) 0.5 keV (b) 1.0 keV(c) 2.0 keV (d) 3.5 keVTotalFocusNoRefArmX+Y effective_area_angle_v10 -1 -2 -1 -2 (i) Fig 6
Effective area as a function of the incident photon angle Θ x with Θ y = 0 ◦ for ID00 at (a) 0.5, (b) 1.0, (c) 2.0,and (d) 3.5 keV. Only the photons collected by the CMOS imager are taken into account. The data points show theresults of the ray tracing simulation, and the curves show the analytic calculation. Point (i) indicates the edge anglewhere the Focus is shfted off the edge of the image sensor. ncident photon angle (deg) E ff e c t i v e a r e a ( c m ) (a) ID20 (b) ID22 id2022_area_angle_v06 TotalFocusNoRefArmX+Y10110 -1 -2
20 22 24 26 2820 22 24 26 28
Fig 7
Effective area as a function of the incident photon angle Θ x for (a) ID20 with Θ y = 0 ◦ and (b) ID22 with Θ y = 19 . ◦ at 0.5 keV. The angles are measured from the optical axis of ID00. The curves of the effective areas aresymmetrical about 19.1 ◦ . Photon energy (keV) E ff e c t i v e a r e a ( c m ) -2 -3 effective_area_energy_v06 Fig 8
Effective area as a function of photon energy for ID00 with an incident angle of Θ x = Θ y = 0 ◦ . xposure (s) flux_limit_v08.pdf F l u x ( e r g / c m / s ) F l u x ( m C r a b ) -11 -10 -9 -8 TotalFocusFocus+ArmTotal (LE)Focus (LE)Focus+Arm (LE)
Fig 9
The 5 σ detection limit of a point source at the center of the ID00 FoV in the 0.5 to 2.0 keV bandpass for thenine-segment MuLE optics. The 5 σ detection limit for the standard LE configuration is overlaid. source was located at the center of the ID00 FoV. To extract foreground and background photons inthe region of ArmX+Y and Focus, we selected the photons in the strip regions along the arm foci(widths of 0.2 cm). The strip width was not optimized but was adequately large enough to collectthe photons focused by the LE segments even in the cases in which the image suffered defocus.Throughout this study, we assumed a Crab-like spectrum for a point source characterized by apower-law photon index of 2.07, normalization of 8.26 photons keV − cm − s − at 1 keV, and anabsorption of N H = 4 . × cm − .The flux limit was governed by the number of photons for shorter exposures (photon limit),and was proportional to t − , where t is the exposure time. Conversely, the flux limit was governedby the CXB photons in the cases of longer exposures (background limit), and was proportional to t − . because the number of background photons obeyed Poisson’s Law.Figure 9 also shows the 5 σ detection limits for the standard LE configuration in which the sizeand properties of the LE segments were exactly the same but the images were not multiplexed.Mathematically, the amount of background was reduced to one-ninth from that of the MuLE con-figuration. The difference between the two configurations only appears in the background limitcase as shown in Fig. 9. Another possible weak point relevant to the nine-segment MuLE configuration is its large FoVwhich causes contamination of bright background sources in the imager. The detection ability of afaint source is easily affected by a bright background source located in any of the nine FoVs. Weconsidered a background point source which was 0.5 ◦ away from the object, which we observed11 onfusion_v04.pdf B.g. source flux (erg/cm /s)100 s10 s10 s 10 -7 -10 -9 -8 B.g. source flux (mCrab)10
100 10 L i m i t fl u x ( e r g / c m / s ) -11 -10 -9 L i m i t fl u x ( m C r a b ) 文字数字:16pt 上付き: 12pt, 6pt上げ Fig 10
Degradation of the 5 σ detection limit in the 0.5 to 2 keV band owing to the background (b.g.) source of the0.5 ◦ separation distance in the case of the nine-segment MuLE optics. at the center of FoV to evaluate its effects. We calculated the detection limit change owing to thebright object for the case of Focus + ArmX/Y described in Fig. 9.Figure 10 shows the detection limits for the observation times of 100, 10 , and 10 s. When thebrightness of the background point source was brighter than 100 mCrab, the detection limit wasdegraded. This is because the flux limit was governed by CXB, which is almost equivalent to a100 mCrab source.There are ∼
30 objects in the entire sky that are brighter than 100 mCrab in the X-ray band.For the nine-segment MuLE configuration that we considered, the FoV was about 666 deg (nine . ◦ × . ◦ FoVs), which corresponds to 1.6% of the entire sky and contains ∼ To evaluate the power of the FoV discrimination by the cross-like image response, we also em-ployed the ray tracing simulation. This problem is converged to a problem that pertained to thedetermination of the azimuthal rotation angle of the cross-like image.
We considered the nine-segment MuLE configuration shown in Fig. 3. When the flux from atransient object exceeds the detection limit, at least one image is captured. At this moment, it isunclear which LE segment (ID00- φ
00 to ID2 - φ
80) focused the image. In consideration of all12ossibilities, the image is subjected to nine different operations to identify the LE segment thatwas involved. The procedure that we employed is as follows.1). For LE segments other than ID00, image distortion should be corrected first given that theimager was tilted with respect to the tangential plane at the center of the LE segment. Thedistortion correction produced eight different images. Details of the correction are describedin Appendix D. Currently, there are a total of nine images.2). By identifying the center of gravity of the entire photons, the position of the transient sourceon the imager O i ( i = 1 , ..., ) is determined in all nine images.3). The position of each photon P i,m ( m = 1 , ..., N ) is recorded, where N is the number oftotal photons. Then, the azimuthal rotation angles φ i,m of the vector from O i to P i,m arecalculated. The azimuthal rotation angles φ i,m are measured from the azimuthal rotationangle φ i of the square cells of the LE segment.4). The azimuthal rotation angles φ i,m are filled in a histogram between − and +45 ◦ giventhat the cross-like point source image has four-fold rotational symmetry. Only the photonsin a ring region of the radius between 0.15 and 3 cm are sampled concentrically around O i .Figure 11 shows an example of the histogram for the case of ID00- φ
00. Herein, there are atotal of nine histograms.5). The point source responses prepared in advance for all nine LE segments are fitted to ahistogram, and the goodness-of-fit was found based on the maximum likelihood estimation.The point source response is generated by the ray tracing simulation with sufficient statisticsfor more than 100,000 photons: CXB photons are not included. The response was modeledwith a Lorentzian function and a constant as according to f ( x ) = S Γ / φ − φ i ) + (Γ / + N. (1)The parameter φ i was fixed to the azimuthal rotation angle of the LE segment cells, and thehalf-width was fixed to the value Γ / . ◦ derived from the simulation. The other twoparameters, Lorentzian normalization S and the constant value N , were free in the fit. Anexample of the fit is shown in Fig. 11.6). The operations are performed for all the nine images, and the one with the highest S/N isselected as the LE segment from which the point source originated.
We performed ray tracing simulations for a transient with a duration of 100 s to evaluate if we couldlocalize its position as a function of the source flux. In this study, the number of CXB photons wasfixed for 100 s observation, but the number of X-ray photons from the transient source was varied.Using the method described in §
40 -20 0 4020Azimuthal angle of photon position φ (deg)6080100 N u m b e r o f p h o t o n s / d e g arm_phase_angle_v05.pdf40 Fig 11 (Histogram) Typical distribution of the azimuthal angles of photon positions for the 250 source and 100 s CXBphotons for the nine-segment MuLE optics. (Curve) The best fit result of the plotted distribution with the templateresponse. average of 350 to 850 trials. Corresponding error bars are also plotted. For simplicity, this studywas conducted with 0.6 keV photons.The number of photons required to achieve 95%, 99%, and 99.7% correct outcome rates forID00 were 142, 182, and 212, respectively. In combination with Fig. 9, the position of a pointsource was correctly determined in 97% of the events at the 5 σ detection threshold for 100 s ob-servations. Even if the correct LE segment could not be determined from the data, the sourceposition on the imager was determined with an FWHM accuracy of ∼
10 arcmin and can be nar-rowed down to nine points in the sky coordinate system. Furthermore, if the correct LE segmentcan be identified with a deep learning approach, the determination accuracy may be improved.Similarly, the number of photons required to achieve 95%, 99%, and 99.7% of correct outcomerates for ID22 were 172, 226, and 266, respectively. The reason for which the correct outcomerate being lower than that of ID00 at the same photon numbers is that the image sensor of ID22was tilted with respect to the tangential surface of the LE segment and the arm of the cross-likeresponse was blurred owing to the defocus effect. It would be useful to perform a more detailedsurvey to assess the performance of the MuLE optics. However, this is beyond the scope of thisstudy and will be described in our next publication.
This study described the working principle of the MuLE optics in which multiple LE segmentsfocused X-rays onto a single imager. This configuration reduced the number of image sensorsconsiderably and thus overcame a disadvantage of the LE optics. A ray tracing simulation wasperformed to evaluate the properties of the MuLE optics based on the assumption of a nine-segmentconfiguration. In the simulation, only the existing technologies (Si–MEMS mirrors and a CMOSimage sensor) that will help with the construction of an inexpensive and accurate enough wide-fieldX-ray monitor in the near future were assumed.14
20 Flux (mCrab)8 10 12200 240 280Number of source photons F r a c t i o n o f s e l e c t i n g c o rr e c t L E m o d u l e fov_determination_v06.pdf
14 16
ID00ID22
Fig 12
Fraction associated with the selection of the correct LE segment for 0.6 keV X-rays detected with the nine-segment MuLE optics.
When the focal length of 30 cm and an area spanning 9 × of an LE segment were used,the total effective area at 1 keV was calculated to be 8 cm at the center of the FoV, and about4 cm at the edge of the FoV ( Θ = ± . ◦ ). The 5 σ detection limit in the 0.5- to 2-keV band for atransient with a duration of 100 s at the center of FoV was ∼ × − erg cm − s − (10 mCrab).The ability to determine the correct position achieved a 99.7% level for a 14 to 17 mCrab pointsource with a duration of 100 s. Thus, we finally conclude that the MuLE optics can be used toimplement a wide FoV transient monitor with sufficient sensitivity.Given that the MuLE configuration is an easiest way to reduce considerably the number ofimage sensors, it is considered to be effective for a small satellite with limited resources or a smallobservatory on-board the International Space Station. With the use of the three units of the nine-segment MuLE with f = 30 cm, as presented in this study, it is possible to cover a 0.75 sr of anFoV with a microsatellite with a volume of 50 × ×
50 cm . With 16 satellite sets, the entiresky can be covered. Using lightweight and inexpensive Si–MEMS technology and by reducingthe number of imaging devices with MuLE, the price per MuLE unit can be reduced considerably.Accordingly, the establishment of a constellation of these types of microsatellites is possible.The ability to cover the entire sky at all times with the satellite constellation will have a majorimpact in the multimessenger and taime-domain astronomy. If the focal length is reduced by halfto 15 cm, the number of satellites in the constellation can be reduced to four, though the sensitivitywill drop. In addition, given that the MuLE configuration that we described in this study canachieve about 1 mCrab at 10 s, it can be used as an all-sky monitor, such as MAXI or RXTE/ASM .Since the position is known in advance, for a known source, it is not necessary to identify the15zimuthal rotation angle of the cross-like image, and the point source can be determined usingonly the location on the image sensor. By optimizing the parameters, such as the increase of thethickness of the Si–MEMS mirror, we can fabricate more sensitive all-sky monitors. In a futurepublication, we will discuss parameter optimization and examine the detailed performance of thoseconfigurations.
Appendix A: Boundaries in the lobster-eye optics
Since the LE segments and LE hollow cells have a finite size, various boundaries appear in theLE optics. Here, we explain the origins of some important boundaries. For the specific numericalvalues shown in this section, the same parameters used in the simulation were used. Figure 13ashows the definition of the LE FoV. It is defined that the center of the cross-like image is exactlyon the line connecting the edge of the LE segment and the center of curvature. With the parametersused in our simulation, FoV becomes Θ FOV ∼ L e /R rad = 8 . ◦ . Figure 13b shows the boundarylimited by the LE hollow cells for the photons that pass through without reflection. This is theboundary visible in Fig. 5a. With our LE parameter, the limit angle becomes θ lim = tan − ( w/(cid:96) ) =3 . ◦ . Figure 13c shows the boundary limited by the support structure (frame) of the LE segmentfor the photons that pass through without reflection. There are no non-reflected photons outside theboundary as seen in Fig. 5b. As observed from Fig. 13c, the location of this boundary is a functionof the incident photon angle. Appendix B: Analytic estimation of the effective area
We summarize herein the methodology to calculate the effective areas. When X-ray photons enter acell of an LE segment, some of them go through the cell without reflection; the others are reflectedby the wall of the cell once, twice, or more times. These photons can be categorized by the numberof reflections. The fraction of each category is a function of the tilt angle of a cell θ j , where j denotes x or y . In the case for which there are no reflections, the fraction is f i ( θ j ) = (cid:40) − (cid:96)w tan( θ j ) θ j ≤ tan − ( w(cid:96) )0 θ j > tan − ( w(cid:96) ) (2)In the case for which there is a single reflection f j ( θ j ) = (cid:96)w tan( θ j ) θ j ≤ tan − ( w(cid:96) )2 − (cid:96)w tan( θ j ) tan − ( w(cid:96) ) < θ j ≤ tan − ( w(cid:96) )0 θ j > tan − ( w(cid:96) ) (3)In our setup shown in Table 1, the boundary angles are tan − ( w/(cid:96) ) =3.81 ◦ and tan − (2 w/(cid:96) ) =7.59 ◦ .Using the photon fraction sorted by the number of reflections, the effective areas are derived as16 e_boundaries_v06.pdf (c) Boundary by the edge of LE segment θ lim
LE cells (b)
Boundary by LE cells (a)
FocusΘ
FOV L E s e g m e n t FrameFocal planeR/2R
Fig 13 (a) The definition of FoV of the LE optics. (b) The boundary produced by the LE cells and the limit angle θ lim . (c) The boundary produced by the edge of the LE segment. These figures are exaggerated for readability. Theboundaries (b) and (c) are common for the NoRef photons and the unfocused direction of the Arm foci. V i g n e tt i n g vignetting_v01FocusNoRefArmX+Y Fig 14
Vignetting curve of the nine-segment MuLE optics as a function of off-axis angle along Θ x with Θ y = 0 for0.5 keV photons. The vignetting is normalized by the value of ArmX+Y at 0 ◦ . follows: A NoRef (Θ x , Θ y ) = AηN x N y (cid:90) θ max x θ min x f x dθ x (cid:90) θ max y θ min y f y dθ y (4) A ArmX ( E, Θ x , Θ y ) = AηN x N y (cid:90) θ min x θ max x f x dθ x (cid:90) θ min y θ max y ξ ( E, θ y ) f y dθ y (5) A ArmY ( E, Θ x , Θ y ) = AηN x N y (cid:90) θ max x θ min x ξ ( E, θ x ) f x dθ x (cid:90) θ min y θ max y f y dθ y (6) A Focus ( E, Θ x , Θ y ) = AηN x N y (cid:90) θ min x θ max x ξ ( E, θ x ) f x dθ x (cid:90) θ min y θ max y ξ ( E, θ y ) f y dθ y (7)where A is the geometrical area L e × L e , η is the open fraction of the pore, N j is the normalizationfactor (cid:82) dθ j , and ξ ( E, θ j ) is the reflectivity of the platinum-coated LE mirror with a surface rough-ness of 1 nm that refers to the X-ray database of the Lawrence Berkeley National Laboratory. The limit angles θ max j and θ min j are restricted by the edge of the LE segment, including the radialspokes and the CMOS sensor. The limit angles vary as the incident photon angles Θ x and Θ y varybecause the viewing angle of the edge changes. Appendix C: Vignetting of the MuLE optics
We showed the effective areas of the MuLE configuration in Figs. 6 and Figs. 7, but they containthe effect by the finite size of the imaging detector. It is worthwhile to show here the vignettingof the MuLE optics. Figure 14 shows the vignetting curve along Θ x with Θ y = 0 ◦ for 0.5 keVphotons. The second peak centered at 19.1 ◦ was due to the ID20 segment.18 ppendix D: Correction method of elongated images detected with a tilted imager The images focused by any LE segment–except the ID00–are elongated because the focal planeimager is tilted with respect to the true focal plane of each LE segment. To correct the elongatedimages, the following operation should be applied: (cid:18) x (cid:48) y (cid:48) (cid:19) = A − (cid:18) xy (cid:19) , (8)where ( x, y ) is the original position of a photon on an imager, and ( x (cid:48) , y (cid:48) ) is the corrected positionof the photon if the imager is located at the proper focal plane of the LE segment without the tiltangle. The matrices A are defined as A = (cid:18) k
00 1 (cid:19) , (cid:18) k (cid:19) , (cid:18) k (cid:48) +12 k (cid:48) − k (cid:48) − k (cid:48) +12 (cid:19) , and (cid:18) k (cid:48) +12 − k (cid:48) +12 − k (cid:48) +12 k (cid:48) +12 (cid:19) (9)for ID20- φ φ
50, ID02- φ φ
70, ID22- φ φ
60, and ID22- φ φ
80, respectively, where k =1 / cos ( θ t ) and k (cid:48) = 1 / cos ( θ (cid:48) t ) . The tilted angles θ t and θ (cid:48) t are defined as − ( L/ R ) and − ( √ L/ R ) , respectively. Acknowledgments
This work was partially supported by the JSPS KAKENHI (Grant Number JP18K18775), TorayScience Foundation, and the budget for basic R&D onboard equipment for future space sciencemissions by the Advisory Committee for Space Science Japan.
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