Design of a high throughput telescope based on scanning off-axis Three-Mirror Anastigmat system
Huiru Ji, Zhengbo Zhu, Hao Tan, Yuefan Shan, Wei Tan, Donglin Ma
aa r X i v : . [ a s t r o - ph . I M ] F e b D ESIGN OF A HIGH THROUGHPUT TELESC OPE BASED ONSC ANNING OFF - AXIS T HR EE -M IRROR A NASTIGMAT SYSTEM
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Huiru Ji, Zhengbo Zhu, Hao Tan, Yuefan Shan, Wei Tan, Donglin Ma ∗ School of Optical and Electronic Information and Wuhan National Laboratory of Opto-electronicsHuazhong University of Science and TechnologyWuhan 430074, China [email protected]
February 22, 2021 A BSTRACT
High throughput optical system is defined to possess the features of both large field of view (FOV)and high resolution. However, it is full of challenge to design such a telescope with the two conflict-ing specifications at the same time. In this paper, we propose a method to design a high throughputtelescope based on the classical off-axis Three-Mirror Anastigmat (TMA) configuration by intro-ducing a scanning mechanism. We derive the optimum initial design for the TMA system with noprimary aberrations through characteristic ray tracing. During the design process, a real exit pupilis necessitated to accommodate the scanning mirror. By gradually increasing the system’s FOVduring the optimization procedure, we finally obtained a high throughput telescope design with anF-number of 6, a FOV of 60 ◦ ×1.5 ◦ , and a long focal length of 876mm. In addition, the toleranceanalysis is also conducted to demonstrate the instrumentation feasibility. We believe that this kindof large rectangle FOV telescope with high resolution has broad future applications in the opticalremote sensing field. K eywords High Throughput Telescope · Off-axis TMA · Scanning Mechanism · Large FOV · High Resolution
Driven by the rapid development of remote sensing, the goal of optical systems design in space exploration, astronom-ical imaging, earth science, military application and many other fields, is moving towards large field of view (FOV)and high resolutions. The researchers mainly concern about the high throughput telescopes, such as Sky Mapper WideField Telescope [1], Panoramic Survey Telescope and Rapid Response System (Pan-STARRS) [2], VST Telescope[3], the visible and infrared survey telescope for astronomy (VISTA) [4], the Javalambre Survey Telescope (JST) [5],which are mainly designed for remote sensing while maintaining high image quality. However, as limited by the sizeof the detectors and the structure of the optical systems, the FOV of these telescopes is usually very small as it iscontradictory to long-distance imaging. Taking the famous Large Synoptic Survey Telescope (LSST) as an example,this large coaxial Three-Mirror Anastigmat (TMA) telescope is composed of an 8.4 meters primary mirror, a 3.4 me-ters secondary mirror and a 5 meters tertiary mirror, has a FOV of 3.5 ◦ [6]. Correspondingly, the large FOV systemsusually cannot achieve long-distance imaging as well [7].In the past few decades, a great deal of researches have been done to balance this contradiction and have made someachievements. As a promising alternative, off-axis TMA systems have attracted the researchers’ attention because ofthe incomparable advantages of compactness, no central obstruction, high degree of freedoms and free of chromaticaberration. These features have great potential to provide good optical solutions to meet the requirements of highresolution while maintaining a wide FOV [8, 9, 10]. The widely applied off-axis configurations usually make the ∗ Shenzhen Huazhong University of Science and Technology, Shenzhen 518057, China
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22, 2021aperture stop offset [11] or FOV biased [12] to avoid obscuration and achieve large FOV. Consequently, the non-rotationally symmetric aberrations dominated by astigmatism and coma [13] caused by the off-axis structures aredifficult to eliminate and make the manufacturing as well as the alignment a big challenge. Fortunately, with thedevelopment of aberration theory, optical system optimization mechanism and the manufacturing technology, theapplication of freeform surfaces has shown great potential in the off-axis optical systems [14, 15, 16, 17, 18]. This isowing to that the freeform surfaces possess the feature of non-rotational symmetry, which can reduce the asymmetricaberration to a certain extent caused by the off-axis of the system [19, 20].In this paper, we propose a design method for high throughput telescope with TMA configuration based on coaxialSedel aberration theory. The design is aimed to achieve a FOV of 60 ◦ ×1.5 ◦ and make image performance close to thediffraction limit for remote sensing. For the requirement of high resolution, an off-axis TMA structure with a relayimage is adopted as the optical configuration as illustrated in Fig. 1. We place a scanning mirror at the real exit pupilto guide the light rays coming from the different FOVs to the overlapped position on the image plane, thus reducingthe size of the image plane to fit the detector. By adjusting the rotation angle of the scanning mirror, the images ofvarious FOVs detected will be stitched together to achieve a large full FOV. When the scanning mirror is fixed in acertain spatial state, the instantaneous FOV is small which means a high resolution. This working mechanism solvesthe contradiction between large FOV and high quality remote imaging simultaneously.The whole design process begins with deriving the optimum coaxial initial structure with no primary aberrationsthrough ray tracing of two characteristic rays, which is detailed in section 2. In section 3, a progressive optimizationis implemented to gradually increase the optical performance. Specifically, we successively add the off-axis quantityto improve FOV and design a scanning mirror to improve the optical resolution. Then, we have analyzed the toleranceof the obtained optical system to demonstrate the instrumentation feasibility in section 4. Finally, a brief summary isgiven in section 5. Relay image
Figure 1: Schematic diagram of off-axis TMA system with a relay image.
The design purpose and specifications are expressly listed in Table 1. Diffraction limited image quality means that theRMS size of the image spots should be smaller than the system’s Airy disk radius, which is expressed by R = 1 . λD f =1 . λ · F / , (1)2 PREPRINT - F
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22, 2021where D denotes the entrance pupil diameter of the system, f represents the effective focal length (EFL), and F/ isthe F number. For instance, if the working wavelength equals 10 µ m, R is calculated as 73.2 µ m.Table 1: Design SpecificationsF number 6FOV 60 ◦ ×1.5 ◦ Entrance pupil diameter 160mmWavelength range 8 ∼ µ mImage quality Reach diffraction limitDetector pixel size 50 µ m×50 µ mDetector size 640mm×512mmFor remote optical sensing applications, a larger FOV is usually preferred, which means a wider target space, asillustrated in Fig. 2. For a two-dimension case, the target dimension for a specific FOV can be expressed as: d = 2 h · tan FOV2 , (2)where d represents the width of the target that can be observed by the telescope and h stands for the detection distance.So, when h is fixed, an optical system with a wider FOV leads to a larger d and needs a higher optical resolution. Also,this will results in a larger size of the mirrors and increasing the difficulty of manufacturing and alignment. To solvethis problem, some studies have proposed the use of FOV segmentation [21] or the push broom scanner [22] to dividethe large FOV into a superposition of small sub-FOVs. As a rule of thumb, a scanning mirror is also adopted in thispaper to divide the full FOV into several sub-FOVs with the usage of multi-configuration. For each sub-FOV, a highoptical resolution is guaranteed. Telescope h d FOV Target space
Figure 2: Geometrical relationship between FOV and detection range d as well as the detection distance h . For the off-axis TMA systems with a relay image, there are two types of structural designs that are widely adoptedaccording to the optical power distribution of mirrors, namely “concave-convex-concave” systems [23] and “convex-concave-concave” systems[12]. As a rule of thumb, the latter one has a relative stronger ability to correct sphericalaberration, coma, astigmatism and field curvature compared with the former one, and is chosen as the optical configu-ration in this paper.We start the initial design by building a coaxial structure which includes a relay image and a real exit pupil, as shown inFig. 3. The aperture stop of system is behind the prime mirror (PM), and all mirrors are designed as spherical surfaces. t , t , t and t represent the distance between the entrance pupil and PM, the distance between PM and the secondarymirror (SM), the distance between SM and the tertiary mirror (TM), and the distance between TM and the image plane3 PREPRINT - F
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22, 2021respectively. r , r , and r denote the curvature radii of PM, SM, and TM, respectively. We set transmission mediarefractive index as n = n ′ = n = 1 , n ′ = n = n ′ = − . PMSM
TMImage
Plane (cid:40)(cid:81)(cid:87)(cid:85)(cid:68)(cid:81)(cid:70)(cid:72)(cid:3)(cid:51)(cid:88)(cid:83)(cid:76)(cid:79)
Marginal Ray PM SM TMImage PlaneChief Ray (a) (b) ZZ u (cid:20) (cid:32)(cid:19) y (cid:21) y (cid:20) y (cid:22) y (cid:23) (cid:32)(cid:19) t (cid:23) t (cid:21) t (cid:20) t (cid:22) u (cid:22) u (cid:21) u (cid:22) (cid:397) t (cid:22) t (cid:20) t (cid:21) t (cid:23) y (cid:20) u (cid:20) u (cid:21) y (cid:21) u (cid:22) y (cid:22) y (cid:23) u (cid:22) (cid:397) y (cid:19) (cid:32)(cid:19) (cid:40)(cid:81)(cid:87)(cid:85)(cid:68)(cid:81)(cid:70)(cid:72)(cid:3)(cid:51)(cid:88)(cid:83)(cid:76)(cid:79) Figure 3: Ray tracing of initial coaxial TMA structure: (a) trace a marginal ray; (b) trace a chief ray.For the propagation of the marginal ray as illustrated in Fig. 3 (a), u is the paraxial ray’s incident slope with respectto the optical axis, and u ′ is the corresponding exit slope. Marginal rays enter PM in a parallel way with respect to theoptical axis, so u = 0 . y , y , and y represent the heights of the marginal ray on PM, SM, TM in turn. For PM, theoptical power can be determined as: φ = ( n ′ − n ) · c = − r , (3)where c is the curvature of the PM. Similarly, for SM and TM, we have (cid:26) φ = r φ = − r . (4)The relationship between the ray height and the ray slope on different mirrors can be determined by the paraxial raytracing formulas, which are expressed as. n ′ i u ′ i = n i u i − y i φ i y i +1 = y i + u ′ i t ′ i t ′ i = t i +1 u ′ i = u i +1 , (5)where i ( i = 1 , , , represents PM, SM, TM and image plane respectively. The ray tracing calculation method ofthe chief ray is similar to that of the marginal ray, as shown in Fig. 3(b). The extension line of the incident ray passesthrough the center of the entrance pupil, which is recorded as the 0th surface, so y = 0. The ray tracing follows: n ′ i ¯ u ′ i = n i ¯ u i − ¯ y i φ i ¯ y i +1 = ¯ y i + ¯ u ′ i t ′ i t ′ i = t i +1 ¯ u ′ i = ¯ u i +1 . (6)4 PREPRINT - F
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22, 2021According to Seidel aberration theory, in a rotationally symmetric optical system, five monochromatic aberrationsincluding spherical aberration ( S I ), coma ( S II ), astigmatism ( S III ), Petzval curvature of field ( S IV ) and distortion( S V ) can be expressed by the structural parameters of the system: S I = − P A · y · ∆ (cid:0) un (cid:1) S II = − P ¯ AA · y · ∆ (cid:0) un (cid:1) S III = − P A · y · ∆ (cid:0) un (cid:1) S IV = − P H · c · ∆ (cid:0) n (cid:1) S V = − P n A A · y · ∆ (cid:0) un (cid:1) + AA · H · c · ∆ (cid:0) n (cid:1)o , (7)where A represents the Snell invariant of the marginal ray, A denotes the Snell invariant of the chief ray, and H is theLagrange invariant of the system. They can be calculated as: A = n ( yc + u )¯ A = n (¯ yc + ¯ u )∆ (cid:0) un (cid:1) = u ′ n ′ − un ∆ (cid:0) n (cid:1) = n ′ − n H = n ¯ uy − nu ¯ y. (8)In this paper, our main concerns are about the spherical aberration, coma, astigmatism and Petzval curvature of TMA.The distortion can be corrected by the specific image processing algorithm, so it is not considered in the design process.From Eqs. (3) - (8), the Seidel coefficients can be expressed as the functions of y , y and r i ( i =1, 2, 3), t i ( i =1, 2,3, 4) as Eq. (9). It can be seen that rays’ heights y and y in each aberration coefficient only control the scales of theaberrations and do not affect the aberrations distribution. S I = y r [ − r r ( r − t ) (2 r − r + 2 t ) − r r r ( r r − r t − r t + 2 r t + 4 t t ) ( r r − r r + 2 r r − r t − r t + 2 r t + 4 r t + 4 t t ) ] S II = y ¯ y r t [ − ( r + t ) + r r ( r − t ) (2 r − r + 2 t )( r r − r t + r t + 2 r t + 2 t t ) − r r r · ( r r − r t − r t + 2 r t + 4 t t ) ( r r − r r + 2 r r − r t − r t + 2 r t + 4 r t +4 t t )( r r r + r r t − r r t − r r t + r r t + 2 r r t + 2 r r t − r t t − r t t +2 r t t + 2 r t t + 4 r t t + 4 t t t )] S III = y ¯ y r t [ − ( r + t ) + r r ( r − t ) ( r r − r t + r t + 2 r t + 2 t t ) − r r r ( r r − r t − r t + 2 r t + 4 t t ) ( r r r + r r t − r r t − r r t + r r t + 2 r r t + 2 r r t − r t t − r t t + 2 r t t + 2 r t t + 4 r t t + 4 t t t ) ] S IV = y ¯ y r r r t ( r r − r r + r r ) . (9)For a general design process, a specific Seidel coefficient is expected to be zero for different design requirements. Forexample, if a flat image field is desired, we make S IV =0, and then solve the equations to acquire the correspondingrelationship among the parameters. By setting reasonable initial values of y , y , t and t , the structural parametersthat minimize these four aberrations and satisfy y = 0 can be determined, thus obtaining the initial optical structure. For this design, we set the initial values of y = 80mm, y = 127mm, t = -180mm and t = 4000mm to make theSeidel coefficients target to zero. The initial structural parameters of the coaxial system, including the curvature radiiof three mirrors and the distances between them, are derived based on the coaxial system design method describedabove. These parameters are listed in Table 2. All mirrors are initially designed with spherical surfaces with coniccoefficients. The virtual stop, namely the entrance pupil, is located behind of the PM. Since the real exit pupil is theconjugate image of the entrance pupil, we will remove the stop at the entrance pupil and set a scanning mirror at theexit pupil position as the new stop later on. In the following design process, the commercial software ZEMAX [24]will be employed for the further design and optimization process.5 PREPRINT - F
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22, 2021Table 2: Coaxial Initial Structure ParametersSurface Radius(mm) Distance(mm) Conic Size(mm)Stop - -180 - Φ Φ Φ Φ Φ After obtaining the initial coaxial structure, we set the FOV as an off-axis configuration in the y direction to eliminatethe mirror occlusion. Specifically, the field angle of 0.5° in the y direction is rotated 15° counterclockwise and theoptical system is symmetric about the y - z plane. In this way, the FOV in y direction is the range from 15 ◦ to 15.5 ◦ ,and the FOV in x direction is the range from -15 ◦ to 15 ◦ . The obtained optical system with the off-axis FOV is shownin Fig. 4. We introduce a folding plane mirror between TM and the image plane for better visualization of the opticalpath. In the actual optimization process, this folding mirror is not considered. y z y z (a)(b) x zx z Figure 4: The initial off-axis structure: (a) side view; (b) top view.In the subsequent stage of expanding the FOV of system, the image quality is the most critical concern to pay attentionto, followed by the problem of mirror occlusion. When the image quality is optimized to the best state, we expandFOV in x to ± ◦ , while increasing the FOV in y direction as a new range from 15 ◦ to 16 ◦ . At the same time, weset a higher weight coefficient of the margin FOV compared with that of the central FOV to compensate for the pooredge imaging quality. Repeating the above optimization procedure, we finally obtained a phased design with a FOVof 60 ◦ ×1.5 ◦ . The whole expanding process of FOV is shown in Fig. 5. For an optical system with a large FOV, the image plane is always so large that is not conducive to optical observationand system construction. This problem is usually solved by adding a scanning mirror along the path of light propaga-6
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22, 2021 x direction y direction FOV optimization points
Figure 5: Schematic diagram of gradually increasing FOV.tion. However, for the system we obtained above, the exit pupil distance varies greatly with different sub-FOVs. Inother words, different chief rays correspond to different optical paths due to the rotation of the scanning mirror andmakes defocusing occur. Alternatively, the multi-configuration structure also has the potential to solve this problem.However, if the multi-configuration structure approach is adopted before the final optimization, it will result in hugecomplexity in the following optimization process due to the numerous variables.To solve this problem, we propose a novel method to control the exit pupil distance. Since the size of the scanningmirror is much smaller compared with the distance between the image plane and the scanning mirror, the image planecan be set as a curved surface under approximate conditions. We set the radius of curvature of the curved surface tobe equal to the exit pupil distance of the chief ray in the central FOV. So, the exit pupil distance of each FOV can beroughly regarded as equal. Based on this assumption, we set the scanning mirror at the position of the exit pupil, andthe state is fixed temporarily. Now, there is only one configuration with a large FOV. Then, we optimize the surfacesof PM, SM, and TM and the distances between them. After the optimization, we obtain the optical system that meetsthe preset requirements as shown in Fig. 6.Figure 6: Schematic diagram of the image surface as a curved surface.However, it is not appropriate to set the image surface as a curved one in practical application and the caliber of TMreaches 1250 mm and the diameter of the image plane is 860 mm, they are so large that bring a great challenge tothe manufacturing as well as the alignment. So, in the following optimization, we will remove the radius of curvature7
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22, 2021of the image surface and combine the deflection angle of the scanning mirror to redistribute the full FOV to multi-configuration design in the x direction. The scanning mirror with ± ◦ rotation is added to the multi-configurationstructure as listed in Table 3. In the subsequent optimization process, the group of mirrors before the scanning mirroris fixed and does not participate in the optimization. As the entire optical system is symmetrical about the y - z plane interms of mirrors and structural settings, only six half FOVs in the x direction will be analyzed, which are correspondingto six configurations in ZEMAX. For the y direction, the angle of ◦ , . ◦ , ◦ and . ◦ are considered in eachconfiguration. Table 3: Multi-configuration SettingsVariable Config. 1 Config. 2 Config. 3 Config. 4 Config. 5 Config. 6XFIE 1 0° 10° 15° 20° 25° 30°XFIE 2 0° 10° 15° 20° 25° 30°XFIE 3 0° 10° 15° 20° 25° 30°XFIE 4 0° 10° 15° 20° 25° 30°Scanning mirror rotation angle 0° 3° 4.5° 6° 7.5° 9°The approximation in the previous process is indeed beneficial to the optimization process, but the optimized structureis not completely equivalent to the structure obtained by adding multi-configuration at the end, which will inevitablyintroduce residual aberrations. For an industrial application, a deformable mirror can be employed to be installedbehind the scanning mirror to improve the image quality. The final optimized optical system is shown in Fig. 7. To make a relatively compact structure, the apertures of thePM, SM and TM are restricted to rectangles. EFL of this telescope equals 876mm, and other optical properties meetthe initial design specifications as listed in Table 1. The structural parameters are detailed in Table 4 and Table 5. PM,TM and SM are all even-order aspheric surfaces with eighth-order coefficients, and only fourth-order, sixth-order andeighth-order coefficients are used. Table 4: Parameters of Final StructureSurface Radius Distance Conic Size tilted about x PM 803.622mm -165.2630mm 0.687 250mm*110mm ◦ SM 1954.405mm 3207.541mm -0.075 420mm*380mm ◦ TM -1894.321mm -1328.788mm -0.094 600mm*300mm ◦ Coordinate Break Infinity 0mm - - − ◦ Scanning mirror Infinity 0mm 0 Φ ◦ Coordinate Break Infinity 0mm - - ◦ Coordinate Break Infinity 1893.190mm - - − ◦ Image - - - Φ ◦ Table 5: Detailed Even-order Aspheric Coefficients of MirrorsSurface 4th order 4th order 4th orderPM 1.218e-10 -1.817e-16 5.420e-21SM -1.552e-12 -9.185e-19 1.750e-24TM -2.817e-12 4.801e-19 -2.076e-24The standard spot diagrams on the image plane of all configurations are exhibited in Fig. 8. To facilitate the comparisonof the relationship with the Airy disk, the size and position of the Airy disks are also provided. It is obvious that allRMS radii are less than 35 µ m which is much smaller than the Airy disk radius of 59 µ m. As mentioned in Section 1, the off-axis systems are more difficult to manufacture and align than the general coaxialsystems due to the asymmetric aberrations caused by the non-rotationally symmetric elements. To demonstrate the8
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Figure 7: The optical layout of the final obtained telescope. ° ° ° xy ° Stru.1 (cid:541) m ° Stru.2 (cid:541) m ° Stru.3 (cid:541) m ° Stru.4 (cid:541) m ° Stru.5 (cid:541) m ° Stru.6 (cid:541) m Airy radius
Figure 8: The spot diagram of the final obtained telescope.instrumentation feasibility of the obtained telescope, a brief tolerance analysis is provided. The tolerance distributionof each component is listed in Table 6. 9
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22, 2021Table 6: Tolerance Setting in ZEMAXSurface Tolerances ValueAll Radius ±10 µ mThickness ±50 µ mPM Decenter in x ±50 µ mDecenter in y ±50 µ mTilt in x y x ±20 µ mDecenter in y ±20 µ mTilt in x y x ±50 µ mDecenter in y ±50 µ mTilt in x y x ±50 µ mDecenter in y ±50 µ mTilt in x y ◦ to ◦ FOV is poorer than that of the center FOV (0 ◦ ) and the edge FOV (30 ◦ ), which coincides with the result of thestandard spot diagram. Nevertheless, 90% MTF is larger than 0.24, 80% MTF is larger than 0.26 and 50% MTF islarger than 0.29 for all FOV in different structures. As a consequence, the obtained optical system has been proved tohave good instrumentation feasibility.Table 7: 2000 Trails Monte Carlo Tolerance Analysis Probability Results of MTF at 6lp/mmConfigurations 98% 90% 80% 50%Config. 1 0.3170 0.3216 0.3237 0.3272Config. 2 0.2808 0.2991 0.3080 0.3214Config. 3 0.2475 0.2683 0.2805 0.3022Config. 4 0.2207 0.2482 0.2645 0.2907Config. 5 0.2431 0.2734 0.2881 0.3262Config. 6 0.2742 0.2980 0.3089 0.3233 In this paper, a high throughput telescope based on scanning off-axis TMA system has been successfully designed.We provide an innovative solution to obtain a large rectangle full FOV by employing a scanning mirror at the realexit pupil. Combined with the progressive optimization method, we finally obtain a telescope with a large FOV of60 ◦ ×1.5 ◦ , an F number of 6, a focal length of 876mm, and the optical resolution reaches the diffraction limit. We havealso provided a brief tolerance analysis to demonstrate the robustness of the proposed design method. The mirrors arethe aspheric surfaces with eighth-order coefficients in this design. If more complex surfaces are adopted, such as theextended polynomial surfaces, a larger FOV and better image quality would be achieved theoretically. And this willbe researched in the follow-up work. Funding.
National Natural Science Foundation of China (61805088); Science, Technology, and Innovation Com-mission of Shenzhen Municipality (JCYJ20190809100811375); Key Research and Development Program of HubeiProvince (2020BAB121); Fundamental Research Funds for the Central Universities (2019kfyXKJC040); InnovationFund of WNLO.
Disclosures.
The authors declare no conflicts of interest. 10
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MTF value @6 lp/mm(
Monte Carlo 2000 trials) structure 1: 0°structure 2:+10°structure 3:+15°structure 4:+20°structure 5:+25°structure 6:+30°structure 1: 0°structure 2:+10°structure 3:+15°structure 4:+20°structure 5:+25°structure 6:+30° C u m u l a ti v e p r ob a b ilit y Figure 9: Cumulative probability change of MTF at 6 lp/mm
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